Can $u(x) = \sqrt{x_1 x_2} + \sqrt{x_3 x_4}$ be solved by Kuhn–Tucker conditions?
Why do you want to guess that some $x$ is zero? What is the problem of having all four goods strictly positive at the solution?  

In fact, due to the presence of the square root in the first order conditions, the solution looks straightforward, since the inequality cannot hold (and assuming exhaustion of the budget).  



ADDENDUM
Given the f.o.c's provided by the OP, all $x$'s have to be strictly positive at the solution, since otherwise we have division by zero and/or the undetermined form $0/0$.  

This in turn implies that $\partial L/\partial x_i = 0$ at the solution, which now becomes easy to compute.