How to find Vout of the circuit?
For what i can see you made 2 faults

First of all,  you switched from KOhm to Ohm. (they differ a factor 1000) 

Secondly, I dont think that is the right way to compute the current through a resistor-diode-resistor circuit. This is what i think:

Name the voltage just above the 9k resistor X. Now the current that flows through the first resistor is equal to 10v-(X+0,7v)/1kOhm. The current that flows through the last resistor is X/9kOhm. The part of the circuit we look at is a series circuit so there must be an equal amount of current flowing at any place. 

This gives us 10-(X+0.7)/1000=X/9000 &lt;=> 90-9X-0.7*9=X &lt;=> 83.7=10X &lt;=> 8.37v

So yes you got the same result as me but i think you are going to make mistakes when the circuits are bigger because your way of calculating the result is really tricky and doesn't work all the time.