Can you possibly extend your stay in London to five days?
I recently (mid-July) took the chunnel from Paris to London. I spent the first night at the London Tower Bridge. There was lots to explore in the neighborhood the first afternoon - Borough Street Market, Southwark Cathedral, the Globe Theatre, Millennium Bridge, St. Paul's Cathedral, etc.
Then the next morning, I headed to the Tower. From the hotel, it was a short walk and I arrived well before the crowds. I had several hours to explore the Tower before the 2:00 checkout that the hotel gave me.
After checkout, I took a taxi to the Hilton Waldorf for four nights. The location was perfect for theaters. The Open Tour bus stop is just half a block from the hotel entrance. We took the tour bus back to the Tower for the free boat cruise to Greenwich and back. In fact, I ended up going back to the Tower Bridge area for another half day to explore the area more.
I understand that AXON is a great value, but if you can tack on an extra day on a weekend, the Hilton London Tower Bridge paid rate is quite reasonable. In fact, next time, I may do two nights at the Tower Bridge and four at the Waldorf.
In terms of room, breakfast, lounge, etc., both hotels allow you to eat in the restaurant if you prefer a fuller breakfast. I slightly prefer the Waldorf breakfast. In addition to the buffet, I could order smoked haddock and kippers (things that I wanted to try). The Waldorf lounge evening offerings were also a bit better.
I got a King Exec room at the Waldorf and one of the bigger rooms at the Tower Bridge. I've heard that rooms at both hotels vary quite a bit, so make sure you ask for one of the bigger rooms. Service was excellent at both hotels.