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You're going to have to re-install windows, but you can still back up the game files just fine.
Install windows normally on the SSD (call this drive C:\), and after you get all of the drivers and everything set up move all of the files from the existing HDD (for clarification purposes we'll call this one D:\) into a folder called 'backup' or something similar, which will avoid the problems that Aerius mentioned.
Note your save locations for all of the games you want to carry over.
Now what you do is install steam to the HDD. This will be the easiest part because after you get the steam application installed as normally (remember to install it to [D:\Program files] or [D:\Program Files (x86)] respective to if you have a 64bit version of windows or not) you can just drag your old steam folder (which will be in [D:\backup\program files\Steam]) on top of it and have all of your old games installed.
For non-steam games you might be able to just drag the games into your [D:\program files], but this can cause some problems as a lot of games require the use of your system user files and registry. Your user files will be backed up in [D:\backup\users\%your windows username%\appdata] (note that appdata is a hidden file and you'll need to enable hidden folders to view it).
Other than that you can either Drag the game's installation out of [D:\Backup] and throw it into your C:\ drive (in a location respective to where the game was installed) or reinstall the games and simply find where the save data is held. Just remember to get any files associated with the game out of your [D:\backup\users\%your old windows username%\appdata] and move it to a respective folder in your [C:\Users\%your new windows username%\appdata].
After you have that all done for every game you should have a shiny new installation of windows with all of your old favorite games installed, and you can delete the [D:\backup] as soon as you don't need anything from it to free up that valuable hard drive space.
Feel free to ask any questions.