Where did my thought go wrong (electrical circuit )
For the first part I'm assuming 0.05 is Vt * n, which I suppose is close enough.  That part looks fine.
The second part is asking you to think about the fact that diode current has an exponential relationship.  So when the current is x times higher, the voltage is larger by an additive factor for any pair of currents (well above Is).  You found in the first part that 10 times the current increases the voltage by 0.115V.  So if you divide 1mA by 10 to get 0.1mA, you already know the difference between the new voltage and V2.
On a related note, there's a well known trick that Vt / 0.43 (or in this case Vt * n / 0.43) gives the voltage change per decade in current.  Technically it's actually Vt * ln(10).  For Vt = 25.9mV, that's works out to a well known 60mV/dec, or in this problem 0.1151V/dec.