Is that Donny-Crewe? Given the weather at the moment I would strongly advise against a taxi, especially doing a route the taxi driver would not be very familiar with.
I am not sure a taxi can be justified either, especially if you are from York, and the failure occurs near York (with ample opportunity to get back to York) and doing a journey Edinburgh-Edinburgh via London.
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Looks like they've swapped the train IDs around, the Chieftain is showing as 1E15 1150 GLC-KGX starting at EDB at 1401. This train left YRK at 1722, and got stuck between Temple Hirst and Shaftholme Junctions. It has just arrived in DON now at 2041!
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Your best bet is probably 2054 DON - LDS (due 2134).
Then, if you are lucky, 2140 LDS - MAN (platform 16A at Leeds) due 2233.
If not, the 2208 gets to MAN 2305.
The train you need to be on is due to leave MAN at 2246, however the 0016 has a connection of -10 at Preston but the sleeper may be delayed so it's a possibility.