A good read AOTG and thanks for posting.
I’m not surprised that the power sockets don’t always work. East Coast uses ancient HST diesel trains on this route and they work hard, all day every day and cover huge mileages.
In addition they were manufactured in the days when mobile phones, laptops etc were unknown. So the power sockets would be a retrofit rather than something designed and built into the original manufacturing process.
Did your daughter use her Young Person’s Railcard ? It offers one-third off most rail fares in both first and standard class.
In future it would be best to allow one hour for the taxi ride back to Kings Cross. London traffic is unpredictable at any time of the day.