The 1st one is better, but I would suggest "for" and not "during". One small point.... If you say "was keeping" this implies that it did not complete.... he was keeping the book for 10 days but then on the 8th day the book was stolen. If you want to show that he completed the action for the whole 10 days, then say "he kept the book for 10 days."

The first sentence is better. "Was keeping" is the verb. The phrase "during 10 days" describes the verb. You were probably thinking the phrase should be next to the verb because it describes the verb. That makes sense, but it's not correct. English is weird sometimes. If you're not sure about the order, put phrases that describe the verb last. That's not a rule, and not all English sentences do that, but it will help you when you're confused. "During" sounds a little strange, though. "During" can't be used to talk about an amount of time. "During" talks about a particular time: during January, during my vacation, during next week. If you want to use it with "10 days", you need to use an article or a descriptive phrase to make "10 days" more specific: during those 10 days, during 10 days in March, during my favorite 10 days. If you want to talk about an amount of time, use "for 10 days" or "over 10 days".

He kept the book for 10 days. (simple past) He was keeping the book during the 10 days he was with us. (Past progressive) Which one are you talking about?

IMHO, the first one, because standard order of words is: subject + predicate + object + adverbial modifier

Anabelle is 100% correct. I notice we use 'for' with the past 'kept for' but not the (what is it) past progressive. We don't say 'was keeping for'. Was keeping implies a continual action without reference to its beginning or end. FOr instance you might describe why, or how it was to be in that action I was walking because I was happy. I was walking and it felt good. For a limited amount of time, we use the simple past tense: I walked for ten minutes. I walked during finals weeks. "During' is an interesting one because we can use it with a past progressive tense (that's a new one on me, good to leanr the name) to indicate a situation. Just like setting the stage for a story. I was walking to school during finals week because parking was impossible. I showed up for class all wet, every time, until people started to make comments. OK, ok. Please don't select this as the best answer, I was just going off on a tangent. Annabelle got it spot on.