Step 1) Replace the 560ti with this XFX 7850. It performs close to a 570 and is a little cheaper.
Step 2) Find out the details on your TV - what's the max resolution (720p, 1080p, etc)? What is its refresh rate (30Hz, 60Hz, etc)?
The monitor you picked looks reliable enough - LG's not a bad brand. Also, the 965 definitely will limit your performance in those two games, but it shouldn't be too bad (especially if you overclock it). It's not like they'll become unplayable - it'll just run slower than an i5.