With a young horse, much of the above advice is a bit too much for a horse just learning under saddle.
No, she does not need to be more forward - that will push her onto her forehand.
No, she does not need to be ridden in sitting trot - her back is not ready.
No, she does not need to be on a seat aid - she does not yet understand the action of the leg and hand.
No, the horse does not need to "give" from the hand - this is incorrect on any horse.
With a horse just starting, basically all we want is that she goes quietly from the leg, towards the bit. As she develops her balance, we expect the contact, connection over the back and impulsion to improve. Right now, pushing her more forward will only serve to rush the trot, make the gait impure and push her onto her forehand.
I do agree that your stirrups are maybe a hole too long, however the MAJOR issue I see is how much time you spend on the wrong diagonal! You should be posting with her outside shoulder as it means you are out of the tack to allow her inside hind to come underneath her body to carry the weight and keep her centered on a curved line. You can see - she almost appears lame when you are on the wrong posting diagonal because she is not able to balance her weight!
The next thing is the clucking and nagging. These BOTH need to stop IMMEDIATELY. She must learn to go forward from a leg and maintain her speed herself. Use the leg lightly once, then cluck and kick if she does not respond, and finally tap her with the whip until she goes. Once she responds, reward her, until you only need to use a little bit of leg and occasionally a little kick to go.
From there, your body stays quiet, your legs hang, your hands are quiet and resting.
As far as exercises, transitions are good. Not too difficult, stick to walk-trot-walk and walk-halt-walk. Introduce a turn on the forehand - praise EVERY step and try in this. And stay very focused on the rhythm - do NOT rush her walk or trot. Once she is established in her balance and rhythm - THEN add some forward and back transitions in the trot. And finally, canter every day. Even just one half circle each direction - she needs to learn to develop her balance. The canter will take a long time to come balanced - so start now!
They say money doesn't buy happiness -- well happiness doesn't buy horses!