Here's a link to a logic board on ebay for $50 you can purchase, which has been taken out of a 5th gen. iPod. The logic board in the link is the only model that's compatible with your iPod (model A1136), the model number being 820-1975-A. Something you should watch out for. There are also two variants of the same model (both having the same model number), the one in the link being the 32MB RAM variant, and the other one being the 64MB variant.
There is also a logic board of the same model that is from a 5th gen. enhanced, or 5.5th gen., iPod which has a higher audio quality because of the Wolfson DAC sound chip built-in to the logic board. According to the description of the logic board in the link listed ahead, it is a lot rarer than normal logic boards of the same model because of this reason. The link on ebay is here. It costs $42 and has 32MB of RAM.
I have a link to an ebay search with the exact model (logic board), and with item location set to worldwide: 820-1975-A. If you want the 64MB variant, here is the link. It costs $66. Here's another one, which is $78, and another one, which costs $65. All three come from 5th gen. enhanced iPods. All the prices that are shown do not include shipping; shipping costs may not be the same for you as it is for me.
This is the guide that shows how to take it apart.
iPod 5th Generation (Video) Logic Board Replacement
I think this covers everything. Hope this helps!