Finding the current trough resistor in a closed mesh
There may be currents entering/leaving the nodes via the spurs. This doesn't matter, you don't need to find these, just solve the isolated mesh with the information given.


\$\small R_2\$ has \$\small 10\: V\$ across it, with + on the left and - on the right.
The \$\small 5\:V\$ source has + on the right and - on the left, since its voltage is -5V.
The voltage across \$\small R_2\$ is \$\small I_x R_2\$ with + at the bottom and - at the top.


Now you can apply KVL to find \$\small I_x\$.

Don't expect KCL to work for this mesh - you have no way of knowing what currents are flowing through the spurs.

You have the correct answer: \$\small I_x = 5\:mA\$