How many Sorcery Points should this cost? Lots. At least 5, since this metamagic is better than Careful Spell, Empowered Spell and Heightened Spell combined. I'm sure you can see where I'm going with this - it's too powerful to put a realistic cost on it.

Your new metamagic option does 3 things. The first is pretty simple - it multiples damage by 1.25, or 1.5 if it's an AoE spell. I'm going to look at Fireball (a pretty classic damage spell) as a standard example.

A Fireball does 8d6 damage, averaging 28 damage. An Empowered Fireball, assuming a Charisma of 20, does 8d6 with the option to reroll up to 5 dice, for an average of 31.75 damage. A Sharpened Fireball does 8d6 * 1.5 damage, for an average of 42 damage.

At this point, you might be thinking "But Fireball does 8d6 damage, where the entry in the Spell Damage table for a 3rd-level spell is only 6d6!" Unfortunately, the advice in the Spell Damage table bears little or no resemblance to the spells that were actually published.

Note that this is also the case for single-target spells. Disintegrate (which uses a saving throw, not an attack roll) deals 10d6 + 40 (average 75), Empowered Disintegrate deals 10d6 with up to 5 rerolls + 40 (average 78.75), and Sharpened Disintegrate deals (10d6 + 40) * 1.25 (average 93.75).

At any rate, hopefully you can see how Sharpened Spell is considerably more powerful than Empowered Spell.

Moving on, the second thing Sharpened Spell does is change the target making a saving throw into the caster making an attack roll. Ostensibly, this is a loss in damage, since most (though not all) spells that use saving throws deal half damage on a successful save. However, it is far easier to make an attack roll hit than it is to make an enemy fail their saving throw.

Firstly, it's a trivial matter to give yourself advantage on an attack roll; it's an extremely non-trivial matter to give an enemy disadvantage on a saving throw. For example, you have inspiration, True Strike, the Help action, the Hide action, The Wild Magic Sorcerer's Tides of Chaos, and heaps of other things to give yourself advantage on a roll. To give disadvantage, you have Heightened Spell, or Contagion. There's not many, and they require significantly more expensive resources than the advantage options.

Secondly, most powerful monsters (the kind that would be worth sharpening a spell for) have Legendary Resistance, which makes it impossible to make them fail their saving throws, since they can just say they didn't. You can make them burn through their uses of this feature, but if they're smart they'll only use it on your best spells.

Finally, attacks are (by design) easier to hit. Everyone uses attacks, and some characters use them exclusively, so monsters can't be impossible to hit without making some players feel useless. On the other hand, not everyone uses saving throws, and everyone can still function without them. It's difficult to prove this mathematically, but anyone can quickly observe that there are extremely few monsters with AC above 20.

A mid-level sorcerer (let's say level 8) with maxed Charisma only needs to roll 12 or higher to hit an AC of 20. With advantage, which you'd better believe they'll find a way to get on their Sharpened Fireball, that's a 69.75% chance to hit.

The same sorcerer casting a regular Fireball has a save DC of 16. Let's assume the monster isn't great at Dexterity saving throws (but not too bad at them, either) and give it a +3. It has a 60% chance of failing that saving throw, which is pretty good odds for the sorcerer, but not as good as the attack roll. On the other hand, if the monster gets advantage on the saving throw (which there are a lot of ways for it to do), it has a 36% chance of failing. Not so good. And, of course, this is assuming a monster with really good AC, which is more of a worst-case scenario than a usual one.

So that's why Sharpen Spell is also better than Heighten Spell. The third thing Sharpen does is change the spell from an AoE to a single-target. This looks like a drawback, and often will be, but in those situations the sorcerer just won't do it. For the sorcerer to be thinking about Sharpening a spell, they'll have a priority target in mind. This lets them use Fireball on the Big Bad without having to worry about hurting their allies. "But they could just use a single-target spell!" Well, they could. But they have a limited number of spells known, and there are in fact no 3rd-level single-target spells. Hence, this is also better (or at least as good as) Careful Spell, albeit only when you have a priority target.

With all that out of the way, this is a really cool idea. It gives us back the Spellwarp Sniper from 3.5, which was great. Focusing Fireballs into concentrated beams of destruction? Awesome.

I strongly recommend going ahead with this, but removing the damage bonuses. If your players choose to use this, they know they're turning an AoE spell into a single-target spell, and they know they're adding a chance of total failure. They'll use it when those aren't problems, so you don't need to boost it because of those drawbacks.

With that change in place, I'd recommend this metamagic option costing 3 sorcery points, the same as Heightened Spell. Wait, what? I said earlier that it was better than Heightened Spell! Well, it is, but it requires the player spending additional resources to make it that way. It's trivial to give yourself advantage on an attack roll, but it's not free. You have to spend an inspiration point, or use your previous turn to cast True Strike or to Hide, etc.

Specific exploits

Simulacrum

With simulacrum, you have a perfectly loyal schmuck who can take all of those negatives for you. They can chill in their permanent Mordenkainen's magnificent mansion being waited on by 100 servants, while you run around with permanent invulnerability.

Wish

Wish can be used to cast spell permanency in 1 Action by duplicating its effects. This would allow you to make an unwilling target's spell permanent or to make a spell permanent on short notice. I cannot think of a specific exploit this would allow, but the behavior may be unintended. Wish will also waive the component cost, which should be minor anyways at the level you can cast wish.

This could be avoided by raising the level of spell permanency to 9th.

Amulet of Health and a warhorse

With some preparation, you can cast the spell yourself. The most debilitating of the three stresses is the reduction in your Constitution score. The rare magic item Amulet of Health sets your Constitution score to 19, which should override the reduction.

Now that you don't die from a stiff breeze, buy yourself a warhorse. You can cruise around at 120' per round as if nothing happened. Better yet, splurge on twelve warhorses - they tend to evaporate at the first sign of fireball.

Speaking of fireball...

Delayed blast fireball

Delayed blast fireball could theoretically deal infinite damage. It starts at 12d6 damage and increases by 1d6 every six seconds. After waiting one day, it will deal 14412d6 fire damage for an average of 50,442 fire damage on a failed save. I hope the target has Evasion or fire immunity!

If you can pick the field of battle, you turn said field - and anyone in it - into a smoking crater. If you can't, you'll need a way to move the bead to where you want. Delayed blast fireball provides the following option:

If the glowing bead is touched before the interval has expired, the creature touching it must make a Dexterity saving throw. On a failed save, the spell ends immediately, causing the bead to erupt in flame. On a successful save, the creature can throw the bead up to 40 feet. When it strikes a creature or a solid object, the spell ends, and the bead explodes.

As BlueHairedMeerkat points out in the comments, the spell doesn't say you can move the bead, only touch and then throw it. The spell says you can throw it, so I am unsure what happens if you choose to touch the bead, pass the save, and then not throw it. Would telekinesis work? What if you're teleported while holding the bead? Your DM may or may not allow you to carry your nuke around.

If your DM allows you to transport the bead, a character can start by making a Dexterity saving throw against the caster's spell save DC. A 17th level rogue can have a +11 to Dexterity saving throws, which rises to +16 standing next to a paladin with 20 Charisma. Throw in the resistance cantrip and a Ring of Protection and they can carry the bead completely safely.

Specific limitations

Concentration

You cannot cast spell permanency on a spell for which you are maintaining concentration. This is because any spell with a casting time longer than 1 Action requires concentration while you are casting it, which will end any other spell you are concentrating on. (Since the duration becomes "permanent", not "Concentration, permanent", I am assuming that any concentration requirement is lifted once spell permanency is completed.)

This limitation is lifted by using wish to duplicate spell permanency.

Balance

The current incarnation is very overpowered, especially for a 5th-level spell. Even as a 9th-level spell, its power would rival wish.

As Ben Barden says in his answer, this spell is nearly impossible to balance. Either you have enough restrictions that no one will use it, or you don't, and everyone will. As new source material comes out, you would need to evaluate each new spell for further exploits. It may instead be easier to create a series of magic items or long-to-infinite duration versions of existing spells, which could each be balanced separately.