Learn Turkish: O kadar deliksiz uyuyordu ki hiç kımıldamıyordu. - She was sleeping so soundly that she wasn't stirring at all.

This is the Turkish equivalent of 'so [adverb/adjective] that [result]'. Here, o kadar (so / to such a degree) pairs with the adverb deliksiz (soundly), and ki (that) connects it to the result clause: hiç kımıldamıyordu (she wasn't stirring at all).

Does deliksiz literally mean 'without holes'? How does that translate to 'soundly'?

Yes, delik means 'hole' and the -siz suffix means 'without'. In Turkish, a 'sleep without holes' (deliksiz uyku) is a very common idiom for uninterrupted, deep sleep. Here it is used as an adverb meaning 'without waking up'.

deliksiz is an adjective, but it's modifying the verb uyuyordu. Doesn't it need an adverb suffix?

In Turkish, many adjectives can be used directly as adverbs without any extra suffix. Just like you can say hızlı koşuyor (he runs fast), you can say deliksiz uyuyordu (she was sleeping uninterruptedly) without changing the word.

Why are both verbs (uyuyordu and kımıldamıyordu) in the past continuous tense?

The past continuous (-yordu) shows that both actions were ongoing at the same time in the past. She was in the continuous process of sleeping, and during that whole time, she was continuously not stirring.