How do I deal with ideal current and voltage sources in my circuits?
If the sources are ideal, you don't have to put any internal resistance on them. Also, I think you are mistaken in assigning a resistance value at the voltage source. It is true that you have a voltage and a current, but the voltage is constant while the current depends on the circuit.

In order to solve the problem, you correctly found the current on the voltage source, and since the text says that it produces 3.333W, it must be the inverse of the arrow.

Therefore, if you apply the KCL at the upper node, you get that \$I_2 = I_S - I_1\$, but since I1 is negative you have to sum the absolute values. I actually don't now how you got the 23/6 A value.

Since you know the voltage on Is, you can calculate the voltage on the two resistances, and since you know all the currents you can obtain the resistance values.