First, thank you for using one of "Top 50 favorite words that just aren't used enough", "avail".
I have tried to spread news of word, avail, but yet to no avail have I been successful.
(That's not a joke, I really do love the word...)
I was wanting to know, after you uninstalled Skype, did you go to C:\Program Files (x86)\Skype and delete the entire contents of that folder?
And after, did you delete the contents of C:\Users\Your Username\AppData\Roaming\Skype ?
Occasionally (actually, all the time) uninstallers will fail to remove previous settings/configuration files. An installer may look for these same files upon installation to restore old settings, if yours were corrupt it may have caused problems during reinstallation.
I just looked up on Skype's website, this advise follows what they recommend:
https://support.skype.com/en-us/faq/FA10634/how-can-i-completely-uninstall-and-then-reinstall-skype?frompage=search&q=can't+reinstall&fromSearchFirstPage=falseThey go a step further to check for old entries in the registry. You can read their details on how to do this. Or... since this is a support forum, I could tell you:
Windows 7:
Click Start, type regedit. Right click regedit.exe and select "Run as Administrator".
XP:
Click Start, click Run. Type regedit.exe, press enter.
When in Regedit:
Click the Edit menu button and "Find". Enter "Skype" in the search box.
Select the search results (Skype.com says "there can be up to 50".) Right click, delete.