Contradiction in solution for ideal transformer
A shorter version of this would be to replace an ideal 1:1 transformer with wires.
The result is the same: two independent sources in a fight.
An ideal transformer with a coupling constant of unity just reflects the voltage on one port to a scaled version on the other, at the same time that it reflects current.  Because it is infinitely stiff in this regard, if you excite it with an ideal voltage source, you have to treat the other side as an ideal voltage source.
Ditto, if you excite one side with an ideal current source, you must treat the other side as an ideal current source.
When you use ideal components, you need to observe some rules to avoid conflicts like this (or you need to accept infinite currents and/or voltages).  That's just how it is.