"I have to constantly ask and pester to do more 'nursey' things".
The most nursey thing you can do is wash and talk to your patient. You will find that you gain the most knowledge about your patient during that time, than any other. Don't let anyone, ever, tell you that washing, talking, feeding or cleaning your patient isn't 'nursey'.
In Intensive Care, the most technical of all nursing environments, the Registered Nurse washes their own patient at all times. If they require help from another nurse, or care assistant, it is the nurse in charge of the patient who views the back of the patient, or the front, etc., so they can use that time to assess the integrity of the patient's skin and identify any vulnerable areas that are prone to breakdown.
Many a time, as a young care assistant and a student, before I qualified, I have admitted a patient and said 'have you ever had any surgery, sir?' "No, never!", they replied. The next morning, assisting with a wash: "What's this scar all about then, Mr x?" "Oh that would be my Cabbage [CABG - coronary artery bypass graft] in 1979!" Ahh....so you had an operation, then? "Oh yes, dear! Quite a big one at the time."
'Nursey stuff' is stuff that deals with patients. Even in intensive care, looking at the patient tells you more than looking at the machines. The machines just help to guide you as to why your patient looks the way they do.