Although his leg was broken, he managed to get out of the car.
=> In spite of
Yes, “In spite of his broken leg, he managed to get out of the car” is a perfectly valid and accurate rephrasing of the original sentence using the phrase “in spite of.” Both sentences express the same contrasting idea: even though his leg was in a bad state, he was still able to perform the action of getting out of the car.
Choosing between “despite” and “in spite of” is often a matter of style and personal preference. Both are generally interchangeable in most contexts. So, feel free to use whichever feels more natural or appropriate for your specific writing or conversation.
I hope this helps! Let me know if you have any other questions.