NavList:

Josh,

With such a variation in index error and some uncertainty of height of eye, the resulting latitude will be somewhat shaky. I have used a true altitude of Kochab of 31°32.5'. Another difficulty is to be sure that Kochab has the same azimuth as Polaris. But let us assume that the azimuths are identical.

This means that the great circle from zenith to Kochab passes right through Polaris. As the time is known we know the declination and GHA of each star. Thus we can calcultate the distance between Kochab and Polaris, I get 16°34.7'. Now the small triangle pole-Kochab-Polaris is determined and it is possible to calculate the parallactic angle at Kochab, I get 8.6'. 

This angle is also included in the large triangle pole-Kochab-zenith. Now it is easy to calculate the latitude:
sin lat = sin dec · sin alt + cos dec · cos alt · cos(parallactic angle), where dec and alt of course referers to Kochab. I get latitude 47°30.1'.

Remains the longitude. Knowing altitude and declination of Kochab as well as the latitude you could calculate the local hour angle. Here you get two possible answers, as cos(x)=cos(-x). But only one of them will, in the end, give the correct altitude. I get -179°49.2' giving the correct altitude. Then longitude equals local hour angle minus GHA. I get -123°10.3', i.e. westerly longitude. 

Lars