SSD will NOT improve your gaming. Your issue for gaming is NOT the SSD if that is the point, while it will help get 'Windows out of the way' that is for a more ..... better performance system. The "intel quad duo... 550 ti" would still bottleneck things and you won't get the 'bump' your looking for. As your HDD is dying, you have a low OLD system, it is time to look at REPLACEMENT instead.
As I noted in other posts let me lay out your options REALISTICALLY... to game on PC means you NEED TO PAY TO PLAY.
Even if you grabbed a i5, 8GB, 1TB off www.slickdeals.net for $399 (for example http://slickdeals.net/f/7665554-dell-inspiron-desktop-computer-660-intel-i5-8gb-ram-1tb-399-99-staples-com?v=1) you can swap out the PSU in there for yours MAYBE IF YOURS can support higher end video cards (check what connections it has) AND shouldn't be a cheap PSU, as it will not fully push 650W but actually as low as 425W, always stick with at least BRONZE rated PSUs. Any 'common' GPU to meet all your qualifications, would be a Zotac GeForce GTX 970 4GB for $310 (almost the price of the whole PC)
http://pcpartpicker.com/part/zotac-video-card-zt9010110p . You could go cheap Nvidia 750Ti that is a additional $125
http://pcpartpicker.com/part/evga-video-card-02gp43753kr for a Low to Medium Graphics, but won't play newer titles as it is underperforming.
In all cases, any changes like that would also have to ask, what computer is this you have now? A Dell / HP / Emachine / etc.? If so the license of your Windows is ONLY working for THAT PC now, you can't 'move it' to different hardware (Windows actually stops working, and won't repair / reload once it sees different hardware), per the OEM Licensing. That is a additional $179 for a new copy of Windows ON TOP of all that.
So the alternative if your NOT able to save up around $800+ for a Gaming PC , then go PS4 (console). You get MUCH better performance, and not have to worry about any of this stuff I mentioned PLUS numerous GREAT titles, very sadly, are EXCLUSIVE to the consoles (like The Last of Us: Remastered 1080p @ 60fps [n] - AARRGGGGG!!!)
Alien: Isolation 1080p @ 30fps [n]
Battlefield: Hardline 1080p @ 60fps [n]
Blacklight Retribution 1080p @ 60fps [n]
Call of Duty: Advanced Warfare 1080p @ 60fps(targeted) [n]
Call of Duty: Ghosts 1080p @ 60fps [n]
Dying Light 1080p @ 60fps [n]
Minecraft 1080p @ 60fps [n]
Planetside 2 1080p @ 60fps [n]
Tomb Raider: Definitive Edition 1080p @ 60fps [n] and so forth.