+1 to that.
Overall, OP, you seem to be of the cheap mindset, just not the super resourceful cheap variety (at least with computers). It does sound like you just broke a small part, and that could very well be replaceable, either by the aforementioned TV repair shop, a resourceful friend, or by a Mac repair shop (most expensive option). Try to get it fixed before you downgrade to an eMac. You will thank yourself later. Oh, and electronics repair 101: don't throw anything away. Those cards you tossed were still worth something. Craigslist, garage sales, and school computer sales (colleges, ideally) are the way to go though if you do get something new.
And have you considered a PC, man? They are MUCH cheaper for getting some decent hardware. I know you may like Macs, but don't force yourself into something that will be too slow for you just because you like OSX... XP, Vista, and eventually Win7 are all decent OS's by now. Something like a netbook or budget tower will blow the socks off of ANY eMac you get...
Oh I'm sure it could be repaired, but I'm not sure it's worth it. I will look into it, but having to replace so much stuff after the logic board is fixed sounds very costly to me; besides that, my ability to put things back together is very limited, in that it just doesn't happen the way it should. While that's probably the best option, it's just not very feasible to me.
Again, speed really is not an issue to me. I use my 1.3 GHz iBook G4 as my main machine currently, and I have no problems with its speed. I understand your concern, but believe me, I am very used to lower end machines.
I do have a decent PC, it's a very good machine in fact. I know it sounds a little fanboy-ish, but while XP is a very decent OS, I can't use it everyday for as long as I would be. OS X has spoiled me.
I'm also not completely decided on this; I might just get my iBook fixed and then save up for a while until I can afford a desktop without having to sell it. Just thinking that this would be a good option.