Some thoughts, having done similar in 2014:
Try this Premier Inn in London, "Premier Inn London City - Old Street" two rooms are about 75 pounds each if you prepay/no changes. You will have to put an adult in each room.
Charming is not what you want, I remember looking at charming B&Bs for 5 in York and they started at 350 pounds.
If money is this tight, consider renting a car at Heathrow, keeping it in York, then dumping it on the north side of London (Luton, Stevenage, St Albans) and taking the train in. You will want something like a Passat or Vauxhall Insignia with an automatic. It's all freeway Heathrow to just outside York, no tolls. York has easy shuttle bus lots that ring the city and for 5 pounds you can get an all you can ride family bus card for the day. You may even find a country B&B outside York where 5 is not such an issue. We stayed at the Holiday Inn Express on the north edge of town, early booking 70 pounds per night, free breakfast. They have connecting rooms.
In London for 5 people usually a black cab is cheaper than the tube, and of course much more memorable. Not of course cheaper if you are going more than say 5 miles.
If you are quick you could churn a credit card for free hotel stays. For example, the Chase Bank IHG card only requires a $1000 spend to get 60,000 IHG points, enough for 6 rooms at the Holiday Inn Express in Stevenage (they also have connecting rooms), so that's 3 nights of 2 rooms. Stevenage is about a 25 minute train ride to Kings Cross. British people will say "Horrors, not Stevenage" but it is a new hotel and a perfectly ordinary British town.
Note that when looking at train fares, you will probably get a group discount if you book the 5 of you at once, and another discount if you return the same day.
This works for Paris, great location, 2 normal bedrooms, bathroom and hall, kitchen area not that nice but works for breakfast. http://www.homeaway.com/vacation-rental/p192356vb