20 minutes to have your system formatted, OS installed and have it back to the way you want it with all programs installed, settings how you like and everything in order. Do you just install Windows and use it for the internet and Solitaire?
To do a fresh install, get OS installed, all drivers up to date, frequently used programs installed and basic settings altered (UAC off, boot optimisation etc) takes me a minimum of a hour and that is before I have any settings in TS, games in Steam, server settings back in Filezilla, favourites and homepages in Opera and Chrome...to do all of that with all of the downloads on my 100Mb/s connection would take me a couple of days.
to assume any of the following:
OP has an image of their system as it was right before the issue happened
OP uses a very basic system with just their OS and the features it comes with
OP has all up to date drivers readily available
would be 100% conjecture and would be the only possible way that OP could do a 20 minute install.
What you are saying is the same as being in a war, having a lone enemy soldier in the middle of a populated city and so decide to nuke them and the entire city flattening it because "hey, they can always rebuild afterwards!!". It is complete overkill and ridiculous to even think it is the best solution here.
To OP, take the advice given and monitor what happens. If nothing is being sent out, it was a password issue that you have solved with the advice given. For precautionary measures, I would still do a full scan and follow the procedures given in the first reply by Punk