Suppose we have a finitely presented group $G$ with solvable word problem. (For instance, the command RWSGroup in Magma terminates giving us a finite [but possibly gigantic] rewrite system.)  Is there then an algorithm to determine whether $G$ has an element of order 2?

If so, where is the best place to find such an implementation?

If not, what more needs to assumed?  What standard methods exist to disprove the existence of 2-torsion?