I'm very happy to give you some thoughts on what 21st-cetury Doctor Who books to nudge towards your LotR-appreciating 11-year-old. There are some quite good 20th-century Who books out there as well! But I shall stick to my remit. (I'm assuming that you have already got this year's annual - the Doctor Who annuals have been of consistently decent quality since your 11-year-old was born.)
The three 12th Doctor and Clara novels that came out this year - Deep Time by Trevor Baxendale, Royal Blood by Una McCormack and Big Bang Generation by Gary Russell - look like a good place to start, linked stories by seasoned writers. I am looking forward also to The Legends of Ashildr by Justin Richards and James Goss which comes out next week - James Goss is my favourite current writer of Who books.
If we're going back a bit further, almost all of the short stories in the Time Trips collection published last year, and the Twelve Doctors, Twelve Stories collection published in 2013, are really good introductions to earlier Doctors.
Favourite books from the 21st century featuring New Who Doctors:
Borrowed Time by Naomi Alderman (11th Doctor and Amy)
Dead of Winter by James Goss (11th Doctor, Amy and Rory)
The Girl Who Loved Doctor Who (comic by Paul Cornell)
Beautiful Chaos by Gary Russell (10th Doctor, Donna and Wilf)
Prisoner of the Daleks by Trevor Baxendale (10th Doctor on his own)
Dead Air by James Goss (audiobook read by David Tennant)
Only Human by Gareth Roberts (9th Doctor, Rose and Jack)
Winner Takes All by Jacqueline Rayner (9th Doctor and Rose)
Two more to add, more for you than for your eleven-year-old. First, earlier this year James Goss published the novel adaptation of the 1979 Tom Baker story City of Death, with the 4th Doctor and Romana in Paris. It's a treat. As I said, I rate Goss highly; here he is working with material originally by Douglas Adams. Second, you of all people are likely to appreciate the jokes in The Shakespeare Notebooks, a series of pastiches of various Doctors in various Shakespeare works. All great fun.
I'm posting this in public, so there may well be further suggestions in comments!