I think she is really, really overtired she is awake for far too long. As a guide, at 5 months old you could aim for something like this
5am - feed and resettle back to sleep
7am wake and start the day
9am nap. If she wakes before 1.5hrs, try to resettle using shush-pat and/or PUPD
11am wake and feed
1pm nap as above
3pm wake and feed
5pm nap 30-45 mins
7pm asleep for the night
Try to make sure she is never awake more than 2 hours. After a short (45 mins) nap, you will need to cut that back even more to maybe 1.5hrs max. If you can't get her back to sleep at 5am, then I would have her asleep for the night by 6pm latest.
What do you think?