Generally speaking, the model described in Kim and Wright(2005)paper belongs to the Gaussian 3 factor model[i didn't see any specification in the volatility term, which means the latent factors does not enter the volatility, for different specification of the short rate diffusion see Duffee(2002) and Duffie and Kan(1996), discrete models are refered to Campbell,Lo and MacKinlay(1997) and Backus,Foresi and Telmer(1996,1998)]. And as a result, the short rate[or instantaneous rate in the paper]term premium is a linear/affine function of the factors.

Now the latent factors.The latent factors are obtained using Kalman filter and maximum likelihood estimation[for other estimation methods, see Piazzesi (2003)].You can find many places using Kalman filter and this technique was originally used in engineering and physics.[Kalman filter can be found in Eviews package, see Eviews manual for Kalman filter updating or Chapter 13 in Hamilton's time series analysis].

Once you get through these, all these analytical solutions, due to this tractability enbeded in this model, for forward rates, expected average short rate and term premium are not difficult to grab[see Campbell,Lo and MacKinlay(1997) for those derivations].

hope these can give you some clues in interest rate term structure models.