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We've stayed numerous times at Bailey's off Glouscester Road in Kensington which is right across the road from the subway. We stayed at a Holiday Inn just around the corner from Bailey's when we needed a place last minute. Last month we stayed at the Strand Palace near Covent Garden (across the street from the Savoy) which was a great location.
I have stayed at the Bedford Hotel in the Russel Square area of Central London several times. I like the hotel and the area very much. The link below takes you to their website which advertises 99 pounds per night. At today's exchange rate, that is about $140 per night. Allow for fees, taxes, and exchange fluctuations, and it is just within your budget. The corporation that owns the Bedford has several other hotels in the same area if you want to spend less. Just be careful with low priced hotels over there as the bathroom will be down the hall.
You asked if you should go to Paris by train also. That is hard to say without knowing how much time you will have in London. I would not bother going to Paris unless you will be in London for at least ten days. During that time, there will be plenty for you to do in London and the nearby countryside. If you have time and want a break, don't bother with Paris; go see the countryside, and other towns and cities. Many major UK towns and cities will only be an hour or two away by train.
As James suggests, Russell Square is a wonderful area of London for a tourist. Near some sights, plenty of restaurants right there or within pleasant walking distance, a little Quieter and less expensive than many other equally convenient areas. Especially great area if you like antiques, REAL bookstores (not Barnes and Noble type) with dusty shelves, fine original editions, and that wonderful 'bookstore smell'.
Unless you've got several weeks, I'd also encourage a train trip out to the English countryside instead of a trip to Paris.
I avoid staying in the central tourist districts. USUALLY busy, noisy, expensive, charmless and rarely have that opportunity to meet locals.
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