First thing in the morning, every puppy has to go really badly. When you are outside with her and she is successful, you should be making a fuss. Praise, praise and more praise. If she will not go first thing in the morning, put her back in the crate for maybe 2 minutes, as she is so young. Take her outside again. Once she is successful, praise.
You have to capture the desired behaviour. The puppy has no concept of what you are asking. She should not have been able to go into the closet after being outside and not relieving herself. She should have been contained, especially after the first time. Those are your mistakes, not hers!
Also, when you take her outside to do her business, do not instigate play until AFTER she has done her business. You want to equate going outside with relieving herself. She will learn to play there later.
I know you are trying really hard, and it is hard. Until she is at least 12 weeks, she won't have a clue about what her body is doing. At 12 weeks, you may only receive a hint that she maybe, just possibly, might have to go. It's up to you to spot that hint and be successful getting her outside in time, when she "asks".
Use each accident as a learning tool. If you always have an eye on her, you will start to know the signs. Lastly, always use the same door to exit so she knows that is the place to go and ask.
Hang in there! Last summer for me was arduous as Henry had the misfortune of being sick, so I had to housetrain a pup with constant diarrhea. It worked and he is just past 1 now and I can't remember the last time he had an accident. The time you put in now makes it all worth while to have a dog you can "trust" to not have accidents for the next many, many years you'll share.
Keep us posted.