Though you've long avoided experiencing morning wood in public: the Timberyard's changing all that. This two-floor rustic-chic coffee shop on Old Street will prop up your java consumption w/ wall-to-wall vintage furnishings, including re-purposed trunks & suitcases serving as tables.
Because trunks don't pack too much high-tech functionality, you can pitch up to an actual table w/ it's own iPad, power sockets, free Wi-Fi, and a stay-for-four-hours-even-though-you've-only-had-one-coffee policy.
Keeping you awake through all of this is a range of coffees via traditional horrendously noisy machines, or Chemex filters paired with a unique and "challenging" blend of beans called Jabberwocky (hopefully it's not too slithy).
They also have a full 17 varieties of tea on the go, all served w/ a pot full of precisely heated water (they have an Uberboiler, which is like a hot tap that you can set to a specific degree), and a timer set to a recommended brewing time for when you should take your leaves.
If you favor depressants they're also stocking beer and wine, or if you're just depressed, eat your feelings with homemade croque monsieurs and pop-tarts, which you, on any morning would.