Solving for an unknown current: Mesh Method, or Node Voltage?
Using the mesh or node equations, you should be able to determine each of the currents and voltages in the circuit in terms of \$V_{in}\$ and \$I_{in}\$. This will give you 10 equations of the form 

$$V_j = f_j\left(V_{in},I_{in}\right)$$
and
$$I_k = g_k\left(V_{in},I_{in}\right)$$

Now you say you are given the power consumed by each resistor, which means a set of equations of the form

$$V_nI_n=P_n$$

where the \$P_n\$ are known values. 

This will be more than enough equations to be able to determine \$V_{in}\$ and \$I_{in}\$. All you need to do is pick the most convenient couple of equations and do some algebra.