Feeling like I'm a failure to my toddler

I have a 21 month old daughter. She has a sibling on the way and a half sister that she sees week on/week off that is 5 years older than her.
She was preemie and a very difficult baby. Didn't keep milk then later, food down and to this day has never slept well. Still wakes up at least once a night then is up for the day at 5am.
I love her to bits, but she is so frustrating. She is always whinging. She makes a constant whine noise for about a total of 4-5 hours a day and this has been getting worse since she was about 13 months. I keep her entertained with painting together, we go to the park a lot and she loves the swings but cracks it when we go home even after 1 1/2 hours.huge tantrums and kicking and screaming then I get looks from others like I'm being a mean mum. She has paper and pens and a lot of toys and things to play with but it's never enough, she just goes back to whinging.
My husband is very caring and very much on the soft side with the kids. He works over a hundred hours a week and isn't home a lot and I do the majority of all kid/house stuff.
I told him my concerns about her and his response "She's just a kid."
She gets upset if her scuttlebug wheel gets caught on something, instead of trying to fix it, throws herself off and has a huge tantrum. I don't indulge in her tantrums and I leave the room but she has started to follow me and will throw herself down again and repeats for every other room I go to.
I thought maybe I was just expecting too much from her, but her new daycare teacher that is nowhere near as good as her old teacher (in for 2 days a week) saw me walking in for drop off and said "oh God! How is she today?" I was a little pissed off at that and smiled sarcastically and said "great like she always is!" I then started thinking if someone from childcare is commenting on her behaviour then what is wrong with her?
She is very intelligent (enjoys flash cards and can name animals/objects/numerals to 3, can count past ten, sorts random objects into colours, pretends to read but does "crisp pointing" to each word of text in a book as she makes up the story, she's self-led toilet training, etc) yet she struggles with a very simple puzzle and screams if the piece doesn't fit in or if she can't find her apps on my ipad, throwing it and takes anger out on me.
She is a bully to any child her size or bigger and gets jealous if I even talk to another child at kindy, trying to push them.
She hits me a lot and has started to hit grandma and her sister that she loves.
She goes to bed at 6:30 after a bath then dinner.
She's certainly not on the autistic spectrum as I work closely with autistic children and she's not red flagging to me that way.
I've taken her to the doctor numerous times and to different doctors and she's always fine and even though she tugs on her ear a lot, her ears are always fine at check ups. I've put in a urine test for her and those results should be back on Monday, but I just feel like a failure as a mum.
Does anyone else's toddler act this full on?