Modelling of a diode
Jim's comment gets your main issue. A typical thermal voltage is 0.025 V, not 0.25 V. In any case, an error estimating the thermal voltage (or the junction temperature) will just cause an inversely proportional error in estimating the ideality factor.

The next issue is more subtle: To accurately estimate the saturation current, you'll want to measure the (very small) current in reverse bias.