My question is pretty simple: how can I export an existing blog hosted on Tumblr to Blogger?
Unfortunately, you're largely stuck unless you feel like doing a bunch of data mangling.
Tumblr provide no full-fidelity export process, and don't seem terribly interested in doing so anytime soon. Gina Trapani provides a little background here as well as a link to a backup tool they produced in 2009 that's Mac-only, beta, and which no longer works due to recent API changes.
Given that Blogger has an import format, your likely if not necessarily practical, best bet might be to create a blog over there with a few entries and export it. Then you can examine that file and reverse engineer it into a Tumblr template that'll output the blogger data format. (I haven't been able to dig up actual documentation of the Blogger format.)
There are also some tools that basically scrape your Tumblr site, like the "tumble-log Backup Jammy," but it produces a paginated HTML dump that you'll still need to wrestle into some other format before you can do anything with it.