I got through only part of reducing long division to medium division before I couldn’t understand the nomenclature being used in the algorithms. If I picked up TAOCP from the begging does it help you get to the point of reading this or is there other prerequisites you need.
It probably wouldn't help, and TAOCP from the beginning is too many pages. It would help to read the division algorithm chapter.
I introduce my own notation from medium->small division onwards. If it's any help u'', v'' are the limbs that the division instruction sees (the ones pertinent to qhat) and u',v' are all the lower limbs.