Finds effective resistance in circuit (no voltage source)
The effective resistance is the voltage developed across the terminals in response to a test current being applied

$$ V = I R$$

So it's entirely normal to use a test current source to make a model from which to extract the effective resistance.

You could also use the conductance form

$$ I = G V $$

with an applied voltage source, and then \$R_{eff}=1/G_{eff}\$.

It will probably be easier to solve this circuit using KCL rather than KVL.

If you must use KVL, you can set up your equations in the usual way. But you'll find you need to take a special case for both your meshes. One will have a current just equal to \$i_s\$. Assuming you choose your meshes in the obvious way, the other will have a current of \$\beta i_x\$. You'll have to express \$i_x\$ in terms of the two mesh currents, and work from there (this will essentially lead you back to the KCL equation for the upper node).