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考え得る限りの手は尽くしたが、問題の解決には至らなかった。
What does の手 mean here? Is it something that can be omitted without changing the meaning of the sentence? It seems like it might indicate a person being in a state or condition.
My translation: Although I thought about it as much I could, I was unable to resolve the problem.
First, the noun phrase 考え得る限り means "as much as one can think of" as you have translated. More literally it would be "the limit of what one can obtain by thinking". It acts a bit like an adjective, because it modifies the 手 following it using の.
手 has several meanings, the original being "hand", but here I think it means something along the lines of a way, trick or a technique.
尽くす means to run out of.
When you put them together, one translation could be
I used up all [tricks]{手} I could think of, but I was unable to resolve the problem.
