Sounds to me I'm sorry like an exercise in futility and wasted resources and effort.
I'll play devils advocate and lets say you get 1.5Ah out of each cell. That will give you 15Ah out of the pack.
I have 2 SLA batteries sitting on my desk in front of me I have been using for playing with arduino projects and other things that are 9 ah each.
I just saw a seller on ebay selling 2 yo cells the same size 2 years old taken from a UPS they want $5 ea for. you could wire these 2 batteries together in 2 min and have more power than what you are talking about. Doing it with AA's just seems like too much effort and resources to make it worth while.
I was also looking at several sellers adverting good ( tested) 100ah 12V deep cycle batteries for $100.
Another thing, is your current battery setup NIcad, lead acid or something else? If it is different, how are these cells going to fit in with regards to voltages, Discharge curves, charging and everything else?
How much time is going to go into the pack you want to make? How much resources in the form of wire, solder, power for your iron, holding/mounting the packs, wasted power in testing and then you don't know how long the get up will last before it all becomes a waste.
Yeah, I get you want to utilise a resource but like a lot of recycling, sometimes that is more energy intensive and uses more resources than it's worth.
I think in the grand scheme of things you might be better off to concentrate on something with a more substantial and likely more reliable return before worrying about something that is going to have as little as what you are proposing.