This may be easier than I thought. Assume WLOG that |K| ≥ |L| ≥ |M| with gcd(K,L,M) = 1 as above.
Then the subgroup H ⊂ ℤ3 of integer translations of ℝ3 that preserves the 2-plane P, defined by
P = {(x,y,z) ∊ ℝ3 | Kx + Ly + Mz = 0}
(where 0 denotes 0) is generated by v = (-L,K,0) and w = (0,-M,L):
H = ⟨v, w⟩ ⊂ P
as a subgroup of P, which itself is a subgroup of ℝ3. We can assume the counterclockwise angle in P from v to w is less then pi radians, and this condition will determine the orientation on P.
But the 2-torus in question is the quotient P / H. Therefore the parallelogram generated by v = (-L,K,0) and w = (0,-M,L) is a fundamental domain in P for this group action.
Therefore the parameter 𝜏 is determined a) by cos(angle(1,𝜏)) = cos(angle(v,w)), where
cos(angle(v,w) = -KM / (√(K^2 + L^2) √(L^2 + M^2))
and b) by the condition that |𝜏| = ||v|| / ||w||, where
||v|| / ||w|| = √(K^2+L^2) / √(L^2+M^2).
The above assumes that either division by zero is a valid operation, or else that the case (K,L,M) = (1,0,0) is excluded a priori as trivial.