There aren't very many (if any) ways to increase your speed permanently or in a traditional manner: looks like Bethesda scrapped that entirely. However, there are a few non-convential and temporary means to increase it:
Sprinting
The basic way to increase your speed is to sprint. Since sprinting uses stamina, increasing your stamina, as Arkive mentions, or your stamina regeneration rate will allow you to move faster for longer periods of time:
- Choose Stamina while leveling up
- Use potions and equip armor with the Regenerate Stamina effect
- The Wind Walker perk in the Light Armor tree will increase stamina regeneration while wearing all light armor
- Using the Become Ethereal shout will prevent your stamina from depleting while sprinting, further increasing your sprint time.
You can also sprint with your horse, although it doesn't appear to be any faster than regular sprinting.
Lycanthropy
If you want to run even faster, contract lycanthropy. In beast form, you sprint much, much faster than normal.
Shouts
While not strictly allowing you to run faster or leap higher, there are a few shouts that accomplish nearly the same thing:
- Slow Time will let you—aptly—slow time to allow you to move normally within your frame of reference but much quicker in others.
- Whirlwind Sprint is an extremely fast sprint that almost acts as a "blink" spell or short-range teleport.
Archery
If you happen to be investing in the Archery skill, you can pick up the Ranger perk, which will increase your movement speed while drawing your bow.
Minimizing slowness
If you exceed your carrying capacity, you'll no longer move at normal speed. So—while not technically a speed increase—if you want to ensure you stay at your speed, you'll want to carry less stuff or increase your carrying capacity:
- Choosing Stamina when leveling up increases your capacity by 5 points
- Activating the Steed Stone will increase your capacity by 100 points
- Choosing the Many Pockets perk in the Pickpocket tree will also increase your capacity by 100 points
Additionally, you can negate the weight effects of your armor, thus maximizing your carrying capacity:
- The Conditioning perk in the Heavy Armor tree will negate the weight effects of heavy armor
- The Unhindered perk in the Light Armor tree will negate the weight effects of light armor
- The Steed Stone will negate the weight effects of any armor you wear.
If all else fails and you're still overencumbered, riding a horse will negate the effects of overencumberence while mounted.
Finally, wearing heavy armor comes with a hit to speed. You can mitigate this by wearing light armor, taking the Conditioning perk, or, as Hafax mentions, activating the Steed Stone.
Since you can't use the developer console it might very well be unsolvable, but they might correct it in a new patch.
Few ideas I've got:
Try to become a vampire, when you are standing in sunlight, all your attributes get permanently (until it is night or you are indoors) decreased, the magnitude depends on which stage you are.
There is an effect, accessible only by using alchemy, that permanently lowers health. It is not the damage one, but Ravage should do it (if I understand it correctly). Although I don't really know whether the potions you create with both positive and negative effects are classified as potions or poisons.
Both of these effects are reversible and might very well not be of any good, but you can at least try.
I am sorry I couldn't provide anything better.
Best Answer
If playing on the PC, the mod, SkyTweak can allow you to decrease (or increase) your experience gain rate.
For example, in this screenshot of the SkyTweak settings here, I am setting the One-Handed skill experience gain rate to half of what it is originally:
You might also want to see How to make Skyrim more challenging at higher character levels?