Current and voltage in a transformer
You should consider separating the two cores into two transformers like this: -







simulate this circuit &ndash; Schematic created using CircuitLab

R1 and the two cores looks complicated the way you have it drawn but the only thing linking the two cores is the current flowing through R1 - that's how I read the diagram and that's what I've redrawn.

Because these are 1:1 perfect transformers with no leakage, you can then move R2 from the right of XFMR2 to its primary and discard XFMR2. Now you have a 9 ohm + 6 ohm load (15 ohm) on the secondary of XFMR1 and, by the same technique this load can be assumed to be directly connected to the input supply voltage of 12V.

This gives a power of: -

P = \$\dfrac{12^2}{15}\$ = 9.6 watts

EDIT to include rationalized picture and calculation of magnetization current: -



Although not shown, the 6 ohm load resistor is connected to the final secondary winding and the total referred load is therefore 15 ohms causing a current of 0.8A. This is load current. Primary magnetization current is 0.6A because \$\sqrt{0.8^2 + 0.6^2}\$ = 1A.

A primary mag current of 0.6A implies the primary mag inductive reactance is \$\frac{12}{0.6}=20\Omega\$. Therefore inductance is \$\frac{20}{2\pi\cdot 50} = 0.0637 H\$. From this you can calculate flux.