No mistakes, but your sentence contains three complete sentences: "I stayed up late", "I had to walk", and "I didn't walk". That is a cumbersome structure. Three subjects and three verbs! I like that you are using verbs, but don't carry it to extremes. Pick out the key action and let that be your verb: "Having missed my morning walk, I stayed up late last night to walk around Hoankiem Lake. Even so, I still managed to wake up early in the morning."

Last night, I stayed up late because I had to walk around Hoan Kiem Lake since I didn't walk in the morning. However, I still managed to get up early in the morning.

Last night, I stayed up late because I had to walk around Hoan Kiem Lake since I didn't walk earlier in the morning. Nevertheless, I still managed to get up early in the morning.