Welcome to the forum
My suggestions are:
Walking along Fleet St from City Thameslink
There is a Season to
London Thameslink, which includes City Thameslink, which around 1 mile - so walkable - from Covent Garden. A 1 month* Season from Luton to London Thameslink is £326.40
The
walk looks reasonably straightforward and may take around 20 minutes. Someone who is more familiar with London than me, may be able to advise on that.
Tube from West Hampstead
A 1 month* Season from Luton to West Hampstead Stns is £280.80, and a 1 month* Travelcard Season for Zones 1-2 is £106.00. Total = £386.80, a saving of around £60 per month compared to a Travelcard from Luton to Zones 1-6. This combination would be valid on trains that call at West Hampstead (not valid on trains that run non-stop St Albans - St Pancras not calling at West Hampstead). On the plus side, the trains that call at West Hampstead are less busy from Luton (in fact, most of them start from Luton) so you will be much more likely to get a seat on these trains, than the faster trains.
Walk to the front of the train, as the exit at West Hampstead (
WHP) is near the front, and go up the steps to the road, it is a short
2 minute walk between the Thameslink (FCC) and Jubilee (LU) line platforms. Change at Green Park for Covent Garden. This would avoid changing at King's Cross.
In a recent thread about St Albans to Victoria., a commute that shares some similarities to yours, it was suggested that West Hampstead is a good place to change:-
The West Hampstead option is better than the Vic line - especially with the new freqeuncies on there - the back of the train is always lighter loaded too !
The back of the train from West Hampstead is where you will be boarding in the morning, after having alighted from the front of the FCC train. So, you have much more chance of getting a seat from Luton - West Hampstead and West Hampstead - Green Park, than if you got a fast train Luton - St Pancras, and then got a tube from St Pancras.
You could try the tube from West Hampstead for a month, and try walking from City Thameslink for a month, and see which you prefer. If you can buy an Annual Season, the best time to buy is just before the fares go up in early January, you get around 2 months of travel 'free' (compared to the monthly cost) and an Annual season also acts as a
Gold Card. Some employers assist with the purchase of such tickets by offering loans deducted from your wages, it may be worth asking your employer if they run such a scheme.
* If you work 5 days a week and do not usually travel at weekends, then it rarely makes sense to buy a calendar month season, and instead buy a Season to start on a Monday and finish on a Friday, lasting typically 5 weeks (varying for bank holidays, annual leave etc).