Finding current through a diode
You have to solve the circuit for every possible combination of diode states. Both on, both off, D1 off and D2 on, D1 on and D2 off. After solving each case using KVL or KCL, you check to see if your assumption was correct. If it was, you're done. If it's not then you solve for the next possible scenario.

The reason you are not getting the right answer is your two loop equations are formulated and solved completely independently of each other.

All loop equations must be solved simultaneously since both 12V and 4V supplies could possibly be pushing current through R2, OR one supply might be pushing so much current through R2 that the voltage drop across R2 is high enough that the other diode is never able to be forward biased and conduct.