Former Londoner here. I always, if I have the choice, like to stay in the Bloomsbury area. Admittedly that's because I like to go to the theatre and I can walk from the West End through Covent Garden back to the hotel afterwards. Buses very easy, tube at Russell Square and a Waitrose in the shopping centre at Russell Square for snacks etc.
I have often used the Bedford Hotel or the Tavistock:
www.imperialhotels.co.uk/en/bedford
which are in your budget. Nothing flashy but you will spend most of your days out and about and in my experience these hotels are clean and professional.
I've also stayed at the Grange White Hall hotel twice (booked by a relative who likes it). If you can get a room at the back of the hotel (but not the basement) it is nice and quiet overlooking a garden. It was a bit tired-looking the last time I was there but the location and the staff are good.
I'd suggest that in this price range it's best not to have breakfast in the hotel. Go out and find a cafe, there's one round every corner in this area. In good weather I have breakfast in the cafe in Russell Square and sit outside and watch the world walk through the gardens. Though possibly not in December.
I would urge you to fit in a show (you can get reduced-price tickets at the booth in Leicester Square) and visit Tate Modern or one of the many other world-class art galleries or museums. You can go shopping in your home town or nearest city but you can't always see a stunning musical or world-class art locally.
What sort of food do you like? If you're not foodies I'm not sure I'd suggest doing something really posh.
Kerridge's Bar and Grill in the Corinthia Hotel down by the Embankment has had some nice write-ups. It's up-market but familiar food. They do a good set lunch for £24 for two courses and I'm waiting for an opportunity to try it out myself. Obviously they also do an a la carte menu for lunch and dinner too if you want to really splash out.
www.corinthia.com/en/hotels/london/dining/restaurants/kerridges-bar-grill
Wear good shoes for walking because you'll cover miles.