In ubuntu 10.10 I cannot boot the system. Can this problem be solved without uninstalling and re-installing the OS?

From your description, it is not exactly clear what is going on. the "its killed" comment is interesting, if you can tell us what is killed, and what OS components are missing.

Anyway, you can certainly recover your data, assuming your HDD is physically intact. Burn a liveCD, then boot to it.
https://help.ubuntu.com/community/LiveCD

The procedure requires a few terminal commands to work -- you will need to manually "mount" (make into a part of your filesystem) your WUBI image (file) in order to read its contents. You can do this by following (roughly) the procedure here.

http://kubuntuforums.net/forums/index.php?topic=3113011.0

If you can boot to the liveCD, then run from a terminal (Ctrl+Alt+T, or from applications menu):

sudo fdisk -l
sudo mount

(thats a lower case "ell"). Then we can give you more exact commands.

I booted from my 10.10 live CD, opened a terminal, and ran the commands listed above and this is the output result:

ubuntu@ubuntu:~$ sudo fdisk -l

Disk /dev/sda: 200.0 GB, 200049647616 bytes
255 heads, 63 sectors/track, 24321 cylinders
Units = cylinders of 16065 * 512 = 8225280 bytes
Sector size (logical/physical): 512 bytes / 512 bytes
I/O size (minimum/optimal): 512 bytes / 512 bytes
Disk identifier: 0xe2c2e2c2

   Device Boot Start End Blocks Id System
/dev/sda1 * 1 24320 195350368+ 7 HPFS/NTFS
ubuntu@ubuntu:~$ sudo mount
aufs on / type aufs (rw)
none on /proc type proc (rw,noexec,nosuid,nodev)
none on /sys type sysfs (rw,noexec,nosuid,nodev)
fusectl on /sys/fs/fuse/connections type fusectl (rw)
none on /dev type devtmpfs (rw,mode=0755)
none on /dev/pts type devpts (rw,noexec,nosuid,gid=5,mode=0620)
/dev/sr1 on /cdrom type iso9660 (ro,noatime)
/dev/loop0 on /rofs type squashfs (ro,noatime)
none on /sys/kernel/debug type debugfs (rw)

I hope this gives you the information you need to help me get my computer back to the way it was. Thank you

In the liveCD terminal session, try these commands:

sudo mkdir /win
sudo mount /dev/sda1 /win
sudo mkdir /vdisk
sudo mount -o loop /win/ubuntu/disks/root.disk /vdisk

This will mount the drive -- the above should complete with no errors. Then run:

gksudo nautilus /vdisk/

This will allow you to browse inside your WUBI install, and recover your files. Use this to copy files from your WUBI install to somewhere safe, like an external HDD, a USB key or a network drive.

 Note that copying the files into your home dir for the liveCD is not safe, as you will lose this when you shutdown.

Thank you very much for this info. I ran the commands and did a full backup of the system, then when I rebooted the system without the live CD, it all booted back up without any problems. Although I did not need the back up data I will continue to keep it in case this ever happens again. This not only solved the problem, but the system is running much faster and has not showed any signs of slowing down or freezing up such as it did before it would no longer boot.

Those commands were only meant to allow you to access your data so you could back it up -- it should not have caused the system boot to change. I am unclear why your problem would have magically resolved itself.

Anyway, note that running ubuntu using WUBI will be a little slower than using a proper dual-boot setup, so if you need a little extra performance, then simply partitioning your drive will give you a performance boost, and should make your system a little more robust with respect to booting problems.

For wubi it's a good idea to keep regular backups (this is a good idea anyway) and you should try to avoid manually shutting the computer down as it can corrupt the root.disk: https://wiki.ubuntu.com/WubiGuide#How_to_reboot_cleanly_even_when_the_keyboard.2BAC8-mouse_are_frozen

Refer to the wubi guide also if you wish to migrate to a regular install:
https://wiki.ubuntu.com/WubiGuide#How_do_I_migrate_to_a_real_partition.2C_and.2BAC8-or_get_rid_of_Windows_entirely.3F