What he said.
The question is close to, "How long will it take me to learn a language?" The answer is different for every person and language.
However, here are some thoughts to help you move along at a good clip during your next trip:
It sounds like you might already be doing some of these (hence you post) but here they are anyways.
You've budgeted some private lessons. Great. Make sure to tell the the Ski School Supervisor what your plans for the trip are - "Would like a private lesson every morning (or whatever you budgeted) for the next XX days." This will make them hopefully pick "an" instructor that can work with you each day and build on the previous days work. If the instructor isn't working out for you, ask for a replacement. The more info you give the Supervisor up front, the closer he/she can get to getting an instructor that fits you (what you want/how you learn).
Go to the Fitness and Injury forum here and do the "Drill of the day". These exercises are geared towards getting you fit for skiing. No sense waiting until right before your trip.
Do lots of balance stuff and building of the support muscles. You can turn you body from sloppy power steering to rack and pinion.
Do balance stuff in you fitted boots.
Just as you're willing to dedicate some time and money towards lessons, dedicate some of that to learning prior to your trip. Get some books, become as obsessive about epicski as you can. There is a ton of knowledge here. I've found the trick is know how to ask the question and then decipher the responses (i.e. is L&AirC full of crap or does he usually make sense to me?).
You can search here and get a ton of response on recommendations for books and videos. Some of the authors and producers (?) are here at epic.
Like Incognito said, do drills. I do them all the time. It's to the point they are part of my free skiing. The more you focus on drills up front during yor trip, the more you will enjoy the free skiing later.
Don't forget to rest during your trip. A tired body and mind is prone to injury.
Don't measure your improvement by what trail you can ski but how well you can ski the trail you're on. There is a difference in ability between 'skied a blue trail' and 'skied a green trail'..."well". Surviving a trail and skiing it are two different things.
Start your mornings skiing on easier trails until you warm up and get you "ski" legs back. Do your drills on the easier trails too, until you feel comfortable enough with them to move to more difficult terrain.
There is a thread somewhere here that talks to this (how long to get to XX level). The answer if I remember right is all over the map. A lot of the replies pointed to advancing is based on what you put into it and where you were (not location but physically and experience wise) when you started.
If possible, go skiing prior to your trip. We're fortunate here to have a few smaller mountains so folks can ski without breaking the bank. Get a skiing tune up prior to your trip so on your first day of your trip, you've already dusted off the cobwebs.
And make it fun. Makes it feel like a lot less work if you're enjoying it.