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  ⊂  ℤ<sup>3</sup> of integer translations of ℝ<sup>3</sup> that preserves the 2-plane P, defined by

   P  =  {(x,y,z) ∊ ℝ<sup>3</sup>  |  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 ℝ<sup>3</sup>.  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).