Need Wireless Router Advice

Hi. I need some router advice.

Here's my situation:

I have an IMac (circa late 2007) running Mountain Lion and it's hooked into a cable modem provided by Time Warner.

Just this week, I bought a 4G IPod Touch. I want to wirelessly allow the IPod to access the internet through a wifi channel generated by my Mac. According to the specs on my IMac, it has a built-in Airport Extreme card.

However, I was not able to successfully get the IPod Touch to access the internet through my IMac. It did see the IMac's wifi channel, but couldn’t connect to it.

I followed instructions on a number of webpages and videos online on how to set up the IMac to wirelessly share its internet connection. None of the instructions worked. I've also been talking to Apple on the phone. They haven't figured out the problem.

But I believe the problem, or maybe it's a symptom of the problem, is that the wifi channel is self-generating its address. In the Network pane of Systems Preference, it says that the Wifi channel had a self-generated address and could not access the internet.

I also called Time Warner and the woman said I need to buy a router. I don't know if she's right, because, as a said, my IMac has a built-in Airport card. Maybe the fact that the built-in Airport card isn't working has to do with Time Warner and not my Mac.

Anyway, I've decided to buy a router as a quick way to solve the problem. However, I don't want to spend much money. I'm out of work and I already splurged on the IPod. Also, I don't need anything too powerful. I'll be using the IPod Touch no more than fifteen feet from my IMac. And I'm not connecting any other devices. other than the I-Mac. Also, my download speeds, via Time Warner, are only 13Mps.

So, the reason I'm confused is that I don't understand the different wireless protocols. I just made a bid on ebay for a used, old Airport Extreme base station. It seems to fit my needs -- it has only one port, and it provides speeds up to 54Mps. But it doesn't support 802.11n, only 802.11a and 802.11b.

So, I'm tying to figure out: will 802.11b and 802.11b work fine for my purposes or do I have to buy a router that uses 802.11n?

Thanks.

Sky