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Vd=(kT/q)*ln(Id/Io) or Vd1-Vd2 = (kT/q)*(ln(Id1)-ln(Id2))
Plot IV curve of the diode. X axis is ln(Id). Y axis is Vd. Find slope of this plot where the plot is most linear. It will be equal kT/q. T - temperature in Kelvin, q is electron charge. So you can find k - Boltzman constant.
When doing the above, plot the current over a very wide range and take your two points over the middle current range that avoids the effects of leakage current at low current values and high level injection effects at high currents.
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