To get a grip on how much time Thaumaturgy takes, you'll want to skip to YS p. 268, "Adjudicating Preparation". Also note that if their Lore is sufficient, you can consider the wizard already prepared and it takes no more play time than an Evocation.
The significant unit of time for Thaumatury isn't a minute, or second, or even hour. Thaumaturgy isn't measured in game-world time units, but rather in game-play time units. Hence, the significant unit of time is the scene, and the amount of time it takes is a matter of GM pacing.
In essence, roleplay the preparation. Do they know what the ritual they need to do is? Their character has to actually do those things. Do they not know the ritual they need? Then they have to go find out and gather the knowledge and resources. Play that out as a quick montage, possibly with skill checks.
That's what the options for the player are there for: Invoke Aspects, Make Declarations, Accept or Inflict Consequences, and (for extreme cases) Skip Scenes.
None of those things should be done as a simple roll and move on. Play it. Playing it out is where the time comes from. Do they have a bloody handkerchief of the person they're trying to locate? That's great, but they still have to do all the FATE stuff to make it mechanically relevant, and to make it mechanically relevant they have to make it roleplaying-ly relevant by playing how it matters and how they use it. This is the beating heart of FATE, where the mechanics demand that roleplaying happen before you can use them. Ignoring this makes FATE fall flat. FATE is a game about showing why the mechanics are justified by playing them first, not by just telling the GM what mechanics are being used and skipping straight to their mechanical effect.
For some examples of what this looks like:
Make a declaration to create an Aspect about it. What are they doing with the handkerchief? Are they focusing their will on it, attuning themselves to the substance of the person they're trying to find? Are they using it to create the circle? Are they arranging it "just so" with a few other personal articles to create a "presence" of the person? Are they smearing symbols on a knife they're going to transform into a "compass"? Ask them what they're doing! What they do will determine what they roll to create the Aspect and what the Aspect will be. Don't just give it to them – make them earn it. Without doing that, you'll skip what makes magic in FATE actually fun to use.
Invoke Aspects. Same idea: they have to earn it by playing it. Players can't just say, "I'm invoking Dear to My Heart because I care about her a lot", they have to show it. They're going to have to roleplay some heart-rending stuff if they want that Aspect. It doesn't have to be long – in fact it can be very brief, as brief as a sentence. But it has to be there, or they don't get the Aspect. Put the fiction in place first, and the mechanics that belong will always be obvious.
Accept or Inflict consequences. In Storm Front, Harry didn't just say, "So I'm going to channel the storm now," and then he cast a spell at the demon. No, he agonised over exactly how dangerous that was, and then did it anyway. He didn't just vaguely channel the storm, he got hit by freaking lightning. He scorched and burned and ached. That is what your players need to do when they're accepting a consequence. The example in the book of the nasty ways consequences can be inflicted takes time too: torturing someone, or sacrificing a living victim, aren't instantaneous. The dark wizard is going to have to get a hold of them, bind them, and actually do the deed. You as the GM need to make that a scene, perhaps a pivotal one. That is where the casting time comes from.
Skip Scenes. This is the fall-back. It's actually the least interesting option, because it takes the player out of the scenes the others are part of. They still need to describe what's going on, else they don't get this option. It should never be: "Okay, I skip a scene to get more shifts." Even the book makes this clear: The player has to describe, even if only briefly, what they're doing that makes them have to skip a scene. Research in the library, ransacking their lab for that one component that they need, and so on.
It should be clear now where the time that Thaumaturgy takes comes from. It comes from all the stuff that the wizard has to do to actually pull off a complex spell, all that stuff that needs to be said by the player and done by the PC. It can sometimes be done as a montage: You don't have to describe every step to the library, just a sentence that evokes images of riffling through shelves, poring over books, and burning candles late into the night – stuff that shows time is passing, and why it's passing.
This is also why Thaumaturgy isn't always strictly better than Evocation: if the character has to actually do this stuff, the world isn't always going to wait for them. If they have to 1) declare an aspect, 2) invoke that and a few other aspects, 3) draw the ritual circle, 4) do the actual ritual, then at some point in that multiple-action process the Bad Guys are almost certainly going to have opportunities to cause problems. It might not even be direct interference – action that the player doesn't like might continue elsewhere.
If a PC wizard is casting Thaumaturgy in the middle of period where time is of the essence – an action scene, or when time is running out some other way – take this golden opportunity to make their life very, very interesting while they're distracted. Interact with their preparations and ritual. They run into exactly the person they want to avoid while trying to get to the library. They need to draw a circle but they only have chalk and it's raining – now what do they do instead? Do they shrug and forego the shift they could get, or do they scramble for an alternative way to create the circle?
Make it fun, make it interesting, play it out, and the time that Thaumaturgy takes will happen all by itself without counting it out.
Best Answer
First off, don't forget that there's a Sleep spell in the main book that uses Thaumaturgy (YS 293), which would make a good baseline.
Apart from that, it looks about right. I'm not sure you'd want to have the Aspect be "Terribly Sleepy", since invoking that wouldn't actually put them to sleep. Invoking an Aspect that implies something will happen when invoked (such as say, "Off Balance") doesn't make that happen; it just makes it easier for someone to make it happen. Invoking "Off-Balance" won't automatically knock someone over, you'd still have to roll Fists or Athletics or whatever to actually knock them over; invoking the Aspect would get you the +2/reroll making it much more likely.
It might be better to have the Aspect be "Light Sleeper" or "Catnapping", since that puts them out straight out (which is what you you want), but still lets people invoke to get a bonus to sneaking or whatnot, but at the same time implies that the target can be woken up easily.