Transport and accommodation in London are big expenses. People always say that eating out in London is expensive but that's only true if you insist on eating at fancy places or right in the middle of the tourist hotspots!
I can't help much with accommodation but you could try
https://www.airbnb.co.uk/ and see what's available if you have an area in mind. If you got somewhere with a kitchen you could go highly Mustachian. Grocery shopping is not particularly cheap in US terms but it's definitely still very cost effective compared to eating out.
Don't discard walking as an option. We live in London and while we obviously use the tube and buses as well, we also walk a lot and it's a great way to get around. London is so old and full of little nooks and crannies that every turn can lead somewhere interesting and unexpected. Look down, look up, look around, because there's always something to see.
We also use that most Mustachian forms of transport: the bicycle. While London is certainly no cycling paradise, it IS possible and cheap to get around by bike and in central London the "Boris Bikes" (cycle hire) will cost you £2 for 24 hours including as many trips as you like, as long as you only use a bike for up to 30 minutes EACH time. There are docking stations all over London where they can be taken and returned. We use our own bikes in the area around where we live, but if I'm in central London and need to do a short trip I'll jump on a Boris Bike. I have an annual membership which is great value but not much use for you!
You probably won't be staying right in the centre, so a cheap way I would personally do it would be take a bus into the centre near your first destination (buses cost £1.40 each trip as long as you're using an Oyster), then try to get around just by walking or using the Boris Bikes, then take another bus back in the evening when you go home. Avoiding the tube is the best way to save money on transport and you see a lot more too.
The museums are of course fantastic and often free. Brilliant are the Science Museum, British Museum and the Natural History Museum, but there are many more, including what seems like HUNDREDS of small bizarre museums dedicated to weird topics or people you've never heard of, which are usually not free but cheap and can be fascinating. I'm not sure when you are going... the Imperial War Museum is outstanding but closed until July for refurbishment.
I wouldn't bother with Madame Taussards and such things, although I do like the London Eye as a splurge (not cheap!).
Hang out on the Southbank if the weather is good. It's just nice. And free. Somebody mentioned Borough Market earlier: you can do a nice walk from that market, up the river to the Southbank. Lunch at the market, then a pint at a pub along the way, end up at the London Eye and Westminster right in the thick of things.
The river itself is sadly under-utilised and there are no real free/cheap ways to actually get out on the water. The boat trip down to Greenwich is nice though if you're willing to pay and you get some witty commentary from the guys on the boats about the scenery on the way.
The walks by the London Walks company are great (I've done a couple) but not cheap. You can also download free podcasts of a bunch of walks by a guy called Robert Wright, which I find excellent. A different experience obviously and a bit weird with two people (both playing the MP3 with headphones on) but they are really great and of course don't cost a penny :)
I know you love trains, but getting around by bus between cities is MUCH cheaper. Megabus, for example, has fares from £1.50 and (just looking at their website) you currently won't pay more than £5 for any fare in England/Wales travelling on Wednesdays!)
If the sun is out, have a picnic in one of the parks on the weekend. It's a very London thing to do. You won't be alone :)
I'm not from London originally but it's my favourite city in the world.
