My belief is that this problem is largely the result of manually shutting down your computer when the Wubi install is hanging. But there have been cases where corruption has occurred without manual poweroffs and I've noticed that some files I delete from within Wubi are 'recovered' later by Windows.
But it is important for users to resist the urge to 'hit the restart/power button' when the Wubi install is hanging. For any OS this is a bad idea and can lead to problems. For Wubi installs it can be fatal. Instead refer to the Wubi Guide for other options.
The first thing you'll notice may be Windows running an automatic CHKDSK or when you try to boot Ubuntu you end up at a GRUB> prompt.
Then when you boot Windows and look in the C:\ubuntu\disks directory you'll notice the root.disk is missing. In some cases, the \ubuntu\disks directory is missing completely or is corrupted.
Running chkdsk
Depending on the problem, Windows may have run an automatic chkdsk or you may need to run it manually. It's not always necessary, but it will never hurt, so the first thing is to run it.
Go to My Computer on XP or Computer on Vista/7, right click the drive you installed Ubuntu on e.g. right click on C:, select Properties, select the Tools tab, then under Error-checking click Check now. Select to Automatically fix file system errors or Scan for and attempt recovery of bad sectors (I usually don't use this, but if you hard drive has problems it's a good idea). When the drive you installed on is C: the computer will tell you it has to schedule the scan for the next time you start your computer. Reboot to complete.
Locate recovered files/directories
The first thing to do is to look for the \found.000 folder on the drive in question i.e. C:\found.0000. This is hidden by default and (on my Windows 7 install) I also had to tell Windows not to "Hide protected OS files" just to see it. You may also have to recover from an administrator command prompt on Win7 (see below).
So now you look for your root.disk (or other .disk files) and copy them back to the \ubuntu\disks folder. If the entire \ubuntu\disks folder is missing, you'll likely find a dir0000.chk directory and within that the root.disk, swap.disk and empty \boot\grub folders. Copy this back to \ubuntu renaming the directory to disks.
If you're missing the root.disk but there is no file of that name, it may have been renamed chk0000.chk. Rename this to root.disk and copy back to \ubuntu\disks.
If the corruption was minor, then likely everything will work fine. If the corruption is major Windows may not even recover the root.disk at all.
Win7/Vista command line instructions
Hit the START key, enter CMD, then look above and right click on CMD.exe and select "Run as Administrator", as shown below. From Windows 8, type "CMD" on the Metro page, right click, and then look on the bottom for "Run as administrator.
Check for the hidden recovery directories: dir /a:h
Check each \found.??? directory:
C:\>cd \found.000
C:\found.000>dir
Volume in drive C is OS
Volume Serial Number is B4B7-99A8
Directory of C:\found.000
19/07/2011 02:02 PM 15,000,000,000 file0000.chk
1 File(s) 15,000,000,000 bytes
0 Dir(s) 222,258,069,504 bytes free
C:\found.000>move file0000.chk \ubuntu\disks\root.disk
1 file(s) moved.
Or if the whole \ubuntu\disks folder is missing:
C:\>cd \found.000
C:\found.000>dir
Volume in drive C is OS
Volume Serial Number is B4B7-99A8
Directory of C:\found.000
19/10/2012 04:51 PM <DIR> .
19/10/2012 04:51 PM <DIR> ..
19/07/2011 02:02 PM <DIR> dir0000.chk
0 File(s) 0 bytes
3 Dir(s) 222,258,069,504 bytes free
C:\found.000>dir dir0000.chk
Volume in drive C is OS
Volume Serial Number is B4B7-99A8
Directory of C:\found.000\dir0000.chk
19/10/2012 04:51 PM <DIR> .
19/10/2012 04:51 PM <DIR> ..
24/02/2012 12:22 AM <DIR> boot
06/11/2012 09:28 AM 13,000,000,000 root.disk
15/11/2011 09:28 PM 268,435,456 swap.disk
2 File(s) 13,268,435,456 bytes
3 Dir(s) 127,904,968,704 bytes free
C:\>move dir0000.chk \ubuntu\disks
1 dir(s) moved.
I hope this helps you to recover your files. Remember to backup important data on your Wubi install. There's no reason to install important personal files on a root.disk - you can access them easily on the /host partition.
Updated 2012-11-06: added some more screenshots, enhanced DOS commands
Updated 2012-11-06: added some more screenshots, enhanced DOS commands