A.H.
She sounds overtired to me. At 5 months she can be taking 3 naps/day. Certainly don't wake her. Make sure to take her outside for lots of fresh air play and put her to bed earlier than 9:30. Any bedtime from 6-8 is great...it just sounds like her overtired adrenaline rush is kicking in, hence she's up ready to party it up with you. I read Healthy Sleep Habits, Happy Child by Weissbluth - it's been our guide and my toddler still sleeps like a champ!