Your expectation of the use of deeply as the appropriate -ly adverb in your example is well founded as a modifier for sleep
He slept deeply
meaning that he had a deep sleep.
However, there are a few other things going on:
Dr. Who is a very stylized programme on BBC1 and I believe it is with this aesthetic that the text is coloured. The excerpt is poetic in nature: too long, too still, too deep, too late. That life is full of stationary time: linger, sit, sleep. That by actively being too inactive, we have not risen to the challenges, nor grasped the opportunities, nor fully embraced all the life has to offer. That in this way, life creeps at a petty pace. That when we finally do rouse ourselves, it is too late, and then we exit.
The allusion of the second shadow is that of the Grim Reaper when our time is over. Where as in life, one faces and moves toward the light, at this point we are facing away from the light. How else would we be able to notice the second shadow?
To maintain the cadence of sit/still, sleep/deep only single syllable words are used, deeply would break this poetic cadence. Then also consider if it is really possible to sit too still or sleep too deep? A possible alternative might be
...sit motionlessly or sleep heavily...
but it would not have the same feel or rythm as the original.
In your example, deep is used as an adverb, though it is not usually used in this way. It is because the verb, sleep, is being used in an active sense: wilfully sleeping too much.
Dr. Who is an English programme. Please do remember it is their language.