Given a diode circuit and some of its values, how do I find the temperature?
Obviously you can't ignore the Is temperature dependence or the diode temperature coefficient would be of the opposite sign (let alone the magnitude) of what we know it to be (~-2mV/K). 

But basically you have the answer. 

Vp  = \$ \frac{n KT}{q} (\ln{(\frac {Id1}{Is}}) - \ln{(}{\frac{Id2}{Is}})) = \frac{n KT}{q} \ln{(\frac {Id1}{Id2}}) \$  

dVp/dT = \$\frac{n K}{q} \ln(5) \$= 0.139mV/K so n = 1 (ideality factor)

(if you got a different number, check your calculations here) 

So the temperature T = \$\frac{Vp}{0.139mV/K}\$ (it's simply proportional to absolute temperature of the junctions, assuming they're the same temperature, of course)

eg, If Vp = 50mV then T = 56.6°C