You know you're not far from like 90 better churches, Anglican, Catholic and Orthodox, indeed the Brompton Oratory, which is Catholic, but very good liturgically, is literally right on your way into the HTB complex.
But if you're not ready to become Orthodox, you would really do better to go to St. Martin in the Fields, or for that matter the Sunday services at Westminster Abbey and St. Paul's. I am rather fond of the Savoy Chapel but the vicar is a little strange. I'm somewhat knowledgeable about Anglican parishes in London, as well as a Catholic and Orthodox, having led my fiancée from atheism to Christianity, and I used to live in Greenwich. So if you would like some suggestions from a layman on churches to visit, send me a PM. By the way if you work in the City, that is to say the Square Mile, many of the Anglican churches therein have nice short and elegant midweek services which are greatly edifying. Forget St. Paul's, you can go to St. Stephen Walbrook or St. Mary le Bow or St. Sepulchre and get straight in. Also the illustrious Kings Weighouse Chapel, where my favorite congregationalist minister Rev. John Hunter presided, and wrote Devotional Services, a beautiful, exquisite revision of the Anglican liturgy that represents the high point of Dissenting Protestant worship in the UK, and that featues one of the most lovely invitations to the Eucharist to be found anywhere ("Come not because you must, but because you may, not because you are strong, but because you are weak"), which I would very much like to see adopted for Western Rite Orthodox use, is now a Ukrainian Greek Catholic Church.
London is a veritable paradise when it comes to having tons of churches in a small area. I have to drive ten minutes to an hour in LA depending on which Orthodox Church I want to go to, and if I want choral evensong in an Anglican parish on Sunday Evening my luck has entierely run out. But London provides easy access to all denominations; whatever you need is just a tube ride away. Or a bus ride, if you like the new Routemasters. My fiancée loves riding upstairs and feels frightened in the Tube; I myself am more of a tube guy because the buses are slow, except in the summer when it gets frightfully hot down there. Especially on the Bakerloo Line, which is otherwise my favorite.