I have to go to London on monday. ... The 10:05 train from Leeds to London (the one I need to catch) is £89 for a standard class ticket and £87 for a first class...
A Super Off-Peak Single Leeds - London on this train bought just before departure is £83, a semi-flexible Super Off-Peak Return is £84 and a fully flexible Off-Peak Return is £132.50 [1], so don't pay £89! £87 for a First Class Advance single, on the other hand, is quite reasonable at this notice.
Alternatively, change during the journey and you can save a lot of money:
To arrive in London just 9 minutes later then NXEC's 1005 from Leeds, book this on the East Midlands Trains site:
Leeds to London Terminals via Barnsley.
0905 Leeds - Sheffield 1002 (Standard accommodation only)
1027 Sheffield - London St Pancras 1234
Currently Advance Singles are £32 Standard and £34 First Class.
I am guessing you want to return home between 1600 and 1700 because the £153.50 First Advance fare is currently available at 1610, 1635 and 1703.
If I were making this journey on Monday, I would return via York. Currently available on the NXEC site are London - York Advance Singles:
1630 Kings Cross - York 1833 £37.40 Standard, £87 First Class.
1700 Kings Cross - York 1851 £37.40 Standard, £87 First Class.
Add £10.40 for a York - Leeds Off-Peak Day Single, and you have still saved a significant amount of money.
Personally, I would book a Standard Advance ticket for either the 1630 or 1700, tell the Chief Steward on boarding that I wished to dine, and there is a >90% chance that on departure I would be invited to the First Class dining car. Allowing £30 - £35 for a very good 3-course meal with half a bottle of wine, it's still significantly cheaper than the First Class fare!
Shop around! There's plenty money to be saved.
John
[1] Super Off-Peak Single valid from Leeds M-F 0905 and all later trains. Super Off-Peak Return valid from Leeds M-F 0905 and all later trains, return from London M-F 0929-1457 and after 1859 M-Th, 1815 F. Off-Peak Return valid from Leeds M-F 0805 and all later trains, return any train
from Kings Cross (there are peak restrictions from St Pancras).