Now before you do anything I would Google the IE9 and IE10 download and save them to downloads first - I keep my downloads in a desktop folder called Downloads.
Start > type downloads right click on that highlighted blue box > send to desktop. http://windows.microsoft.com/en-au/internet-explorer/ie-9-worldwide-languages http://windows.microsoft.com/en-au/internet-explorer/ie-10-worldwide-languages
These are the refs for here in Australia you will probably get the site from wherever you are. Now do not use anything but the Windows system of pulling up the regedit
Start > type regedit - it will show the highlighted box see my pic the quality isn't good as you cannot snip it.
It sounds as if you have a few of add on apps to make the machine go better or improve the performance I would urge you to look at some of those and if they do things with the registry like cleaning it and defragging it they need getting rid of as most cause more problems than they are worth. The only exception is the CCleaner app from piriform.
If I were doing this I would then go to regedit and open to the HKEY_USER_MACHINE > Microsoft > Software > Internet Explorer don't open it right click on the whole ref and then delete. It will ask you if you want to delete this entry and all its sub entries.
Go to Downloads and run the IE10 - if that is what you are after and run / install it or you could try that through updates.
If that does not install then you still have the IE9 in the Downloads to do the same with.
AS I have said this is what I would do but the choice is yours as the outcome could be that you have to do a complete reinstall of the OS if things are a bit messed up.