How to Solve TSP Given Length of Path?
This decision problem ("does this graph contain a tour of length at most this?") is NP-complete, so you won't find a polynomial-time algorithm most likely. 

In addition to the brute-force method you describe, you can have a look at Held-Karp, which solves the problem in $O^*(2^n)$ time using dynamic programming. For an even more practical solution, you can have a look at the Concorde TSP solver, which has been successfully used to solve instances with tens of thousands of cities optimally.