Yes, this is a known problem. By setting the compability to Windows 95 it should solve so you can actually start playing, though I woulden't reccommend it as it can make your mouse very choppy/lag in-game.
But I found out how to solve this too. You need "Windows Applications compatibility Tools", I am using version 4.1. It is free to download from Micro$oft, so if you google for it like I did you won't have any problem finding it.
Before you start be sure you don't have changed anything on your *.exe file, such as set the compability to windows 95, if so change it back.
Now install the tool from Micro$oft and start by using "Compatadmin.exe"
1. Click on the "new" tab to make a new database, name it dungeon keeper 1
2. Right click on your new database and choose Create new -> Application fix...
3. Write Dungeon Keeper 1 for name of the program, Bullfrog for the vendor and then click the Browse button to find your dungeon keeper exe file such as "keepd3d.exe" or "Keeper95.exe". Then click next.
4. Choose "none" on the left list and click next.
5. You should have "EmulateHeap" turned on in this list, that is what you need to actually start the levels. You can also turn on "EmulateSlowCPU" although I have yet to see if this actually helps for something. Then click next and finish.
6. Save your database somewhere and then click on the "file" tab and choose "install"
It should now work fine.
P.S. If those that are interested knowing what actually make the mouse laggy it is when you set the compability to windows 95, then it turns on 57 compability things, one is "EmulateHeap", and that is why you can start the game, but it also turns on "DisableBoostThread" which is the source why your mouse is laggy, the other 55 things I am not sure about what they do.
It should also be possible to turn off Hyperthreading in the list, but alas I have forgotten what it is called. So if anyone know please write it here, I have used it for Grim Fandango where Hyperthreading causes crashes.
Anyway, hope this helps and good luck.