Micro ATX motherboard upgrade

Unless it's broken, I wouldn't change out the motherboard. You'd be better off upgrading the power supply and video card, then trying your new cpu. Your board might run one up to 95 watts; also if you're using the original acer windows installation, it won't run with a non acer board. Oem systems are harder to upgrade.

Actually it did break. But being the beginner I am, I have no idea what part of it broke. It doesn't turn on at all so I'm assuming it's the Power Supply. Is there any reliable method of finding out what's broken?

You can use the paper clip test on the power supply, but I've never tried it. If you still want to change the board, change the cpu also in case it's fried and sell the leftover parts. Here's some suggestions:

http://findnsave.star-telegram.com/Local-Ads/a-16883/Fr... (in store only)
or this newegg combo:

http://www.newegg.com/Product/Product.aspx?Item=N82E168...
http://www.newegg.com/Product/Product.aspx?Item=N82E168...
http://www.newegg.com/Product/Product.aspx?Item=N82E168... (one or two sticks, your choice)

I recommend this basic ps if yours fails the paper clip test:
http://www.newegg.com/Product/Product.aspx?Item=N82E168...

Want windows 8 and not do anything else to your setup:
http://www.fatwallet.com/Dell-Consumer-coupons/dell-hom... (may be expired now, but with the i5, it's a pretty good deal)

Sorry, but I was mistaken on the dimension, they're actually reversed, so 7.8" x 9.6"
The side with the IO ports should be no longer than 7.8".
I appreciate the effort, but I guess this case is just really stupid. Would you recommend that I just buy a new one?
Also the strongest Power supply that I was able to find that could fit was
http://www.newegg.com/Product/Product.aspx?Item=N82E168...
as you can see, it's a reaaaally small micro ATX casing.

You may just want to start over and sell off the leftover parts. But here's a board that will work with your old cpu and might fit; you'll still need ddr3:
http://www.biostar.com.tw/app/en/mb/introduction.php?S_... ($44.99 with free shipping at newegg; link broken). Your cpu is listed under "cpu support". The 4 gb stick of ddr3 I linked above should work, but may run at 800 or 1066 speed, which won't hurt the ram. You may also need to call microsoft for a new windows coa number. If yours was installed by acer, then you can purchase a windows coa on craigslist for xp or 7; get an unopened copy.