To be able to visit London on a budget in half term?

Camden Market – still buzzing, free to visit and there’s loads of street food options.

The Delaunay is a fancy restaurant but next door is the Delaunay Counter – a lot of the same, delicious dishes from the same kitchen but for less money

Travelodge in Finchley Central is two minutes walk to the tube and gets you into Central London in 30 minutes and is cheap ( though we went in term time so they may rachet up the prices in school holidays).
It was perfect for us. Hope she has a lovely time.
Covent Garden in the evening is magical, lots of quirky stalls and some really impressive busking, really, really good.

Is this £400 including your £100 ticket on train? So £300 in London?
Are you planning to stay one night?

You can stay further out and take tube in. I'd suggest the Premier Inn at Tottenham Hale station - it's right by the tube station which is on the Victoria Line (will take you to Euston directly as well as Oxford Street, Green Park etc) and it is on fast overground rail to Liverpool street where you can change to Elizabeth Line, central, circle lines etc.
The area around the hotel is not exciting (it's basically new build flats, student dorms and a Homebase) so you'll want to eat dinner while out in town and then head back there just to sleep.

You need to apply at least 28 days in advance but you can save on your daughter's tube and bus fares while in London.
https://tfl.gov.uk/fares/free-and-discounted-travel/11-15-zip-oyster-photocard#on-this-page-0

Teenagers often enjoy browsing quirkier areas like Camden Market? Vintage clothes, music, decor for a bedroom etc. And browsing is free.

Quirkier museums: Sir John Soane's museum. The Hunterian (bit gruesome, it's history of surgery). Denis Severs house.

The Wellcome collection is right by where you arrive at Euston and has interesting explorations of disability that might be of interest?
https://wellcomecollection.org/exhibitions/ZZP8BxAAALeD00jo

Hi- with accommodation, I don’t think you’re going to be able to avoid it being expensive I’m afraid, but do have a look at premier inns or travelodges at places along the main tube lines to see if they are any cheaper ones? for instance there’s one in Richmond which is nice and you can get the tube or train from there into central london and you are also close to the river and Richmond park there too.

For days out, there are many cheap museums and parks : it depends a bit on what your daughter is into.. She may like to spend some time in one of the Westfield shopping centres (Stratford or Shepherds bush). Also, if you book well in advance (try as soon as you can) you can get tickets to see the view from Sky garden . Greenwich is good as you can get the DLR or the Uber boat there, visit the maritime museum for free and then there are good planetarium shows at the observatory for only £6 a ticket. Also try the free galleries too: national, national portrait, Tate and Tate modern.

It should be possible for you to stay further out e.g. zone 3 or 4 and take public transport in. Only because I know the area, there's a Travelodge by Walthamstow Central station, for example, which is likely cheaper than the ones in the centre, and you could take the overground to Liverpool Street or catch the Victoria line from the same station, both of which would get you to the centre of town in 20 mins or so.

Oh and in terms of accommodation, look east to the square mile. My DH stayed in a big hotel room at moorgate last week for £100 a night. Lots of availability in the city esp at the weekend.

And when I say cheap seats usually no more than £30 a seat.

You mention a disabled railcard - Does your daughter have mobility requirement? If so this can impact tube use as only a few tubes have lifts.

If no requirements, then I'd recommend seven dials market for lunch and the whole area around there for shopping, into Covent Garden for sephora etc. Some cool vintage and jewellry shops at 14yr old budget level. Lots of teen girls around there (I have a DD16 who discovered seven dials with me when she was 13 and now goes with friends on her own!)

My DD also loves musical theatre. If you don't mind cheap seats (which are always much better than you think, we only buy cheap seats) then you can get tickets to a show.

Premier inns are much cheaper the further in advance you book. We paid about £112 per night last October for a family of 4 - but we booked it the November before. Have you checked prices for half term in February, or Easter? I would imagine trains may also be cheaper if booked further in advance?

What’s the nature of her disability? London can mean a lot of walking. Buses are great but sometimes slow in traffic or if you are able to get the tube you could look at premier inn/travelodge further out but by a tube stop. Finsbury Park/Archway/Manor House arent the fanciest areas but you can be in central London in a little over 10 mins. Southgate/walthamstow etc a little further.

could you see if there are any cheap tickets for a show? Mean Girls/wicked etc.

Whereabouts are you travelling from? Might be cheaper to get a national express coach rather than a train?

She might like to see a musical? Mean girls, Six, Moulin Rouge?

The sky garden is pretty cool and is free. You can also go to some of the bars in the Shard and just have a drink (as long as you meet the dress code!).

Food wise you could stick to chain places like Bella Italia, wagamamas etc. Could also use the toogoodtogo app to get stuff for breakfast and lunch and have a picnic in a park if the weather is okay.

There was a 2 for 1 entry scheme a few years ago, so if you travelled by train you got one free entry to certain attractions. But you probably have an Access card anyway.

I can't help with accommodation ideas, but I went to Frameless just off Marble Arch recently, and it was brilliant. An interactive art gallery. I keep recommending that to everyone looking for something to do in London.

She might like the V&A museum? Lots of jewellery/gems to look at from what I remember.

The Thames Clippers going east-west or west-east aren't very expensive and it's really nice when you see all the sights lit up at night.