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Stephen,

I didn't use and app. I thought I'd try using Nautical Almanac tables first. My first steps was to worked out a rough idea of latitude. You need the Local Hour Angle of Aries, which means you need a guess as to longitude, to that that very accurately, so I took a gander at the A0 table in the nautical almanac and noticed that the correction would be between about 21' and 48'. So I took a rough average of thsoe and went with that. 

I then on a universal plotting sheet, I drew a course line from the latitude I worked out from the Polaris shot and came up with a latitude for the second shot by advancing my line of position as far as I would go at 5 knots in the period between the two shots on a heading of 225°T. 

Then I noticed that the height of the sun at the time of the shot -- close to 15° -- meant that I must be east of Greenwich, which surprised me. I did a rough calculation that I'd be maybe 23° east of Greenwich -- same as what you got!! -- and did a fix based on that. That put me in the Libyan desert. So I fudged a bit and decided I must have been off in my Polaris sight reduction and that I was on course forDarah. 

The next step would be to get a more precise longitude by working out what longitude I would have to be for the sun to be at 14° 24' at  05:01:30 UTC. Then I'd go back and redo the polaris sight reduction and the sun LL sight reduction with that as my assumed position. I bet I could get a pretty good fix that way.