You have my sympathies- I had something similar when going into labour slightly prematurely at 4am in the morning! Paracetamol (& ibuprofen) do not work - don't bother and don't take it, as it is not the kind of pain the tablets would be effective against. If any consolation I had a TENS machine in the house with NO batteries in it at the time - argh! - as it was "too early to go into labour!" You can last till 7am, trust me.
Warm bath may help - run a warm bath as though you are using it as a hot tub/ jacuzzi / water birth tub, even if you have no plans to have a water birth at all. ie get someone at home to run this for you.
Massaging the base of the back helped me - may work for you.
Distraction - eg music, watching a funny DVD. Don't be perturbed that you are awake while others are asleep- they're not in labour!
Walking around slowly can help. Prior to getting into the car, it was the only thing that helped me as I was getting a bit red/sore from massages, and couldn't actually get myself into a bath tub because of the position of the baby.
You may also find it useful to distract yourself by checking your hospital bag and making sure there is everything you want, and that you don't pack valuables that might be lost in the hospital.
If you are already booked in for an induction for 7.30am, then in reality you can leave earlier to 1) beat rush hour traffic, 2) have fewer people in the entrance/car park etc, so in effect you can start thinking of going at 6-ish onwards (depending on how far you are from the place) - doesn't hurt to be very punctual (wink). They won't turn you away just because you are early, but bear in mind if your labour ward is normally extremely busy it may be more pleasant waiting it out at home.
Obviously if anything changes significantly between now and 7am which worries you, get in touch with the midwives again.
Best wishes to you and your baby! 