Tate Modern is one of my favorite places in London. I've been also planning to go to Tate Britain for a while now, but it is a bit awkward for me to get to from my usual hunting grounds, so I have yet to actually make it happen. Perhaps in June.
Your walk almost copies the one I did last I was in London, except then I spent som time walking about the back alleys between Ludgate Hill and Blackfriars, to end up going to the Cheshire Cheese pub. Actually had a proper fire in a fireplace going, that was nice to have a pint next to. Man I like visiting London.