Hi Cadge
Have you had a look at some of the posts on our NY section, lots of useful info there.
Staying on 43rd/8th you are within easy walking distance of many of the activities around Times Square. You could walk to the Rockefeller Center and go up to the Top of the Rock, which I preferred to the Empire State Building. I went to the top late afternoon and stayed up there until it was dark and watched all the lights come on all over the city, quite magical, though a bit cold.
I felt quite safe walking (on my own) even late at night after going to a show as long as I kept to the main areas. No problems riding the subway, though I didn't use it at night.
Many tourists ride the hop on hop off bus I think there is both a one or two day ticket on offer, it would give you a good idea where everything is. I chose to book a 6hr guided tour of Manhattan with lots of stops, on my first day there. It departed from Time Square around 9am. and we were given the option of being dropped off at Macy's Department Store. at the end of the tour. I would highly recommend the tour. The guide was extremely knowleagable and I was able to find my way around quite easily after the tour. When I was there Macy's were giving a discount to foreign tourists, I don't know whether it's still available but worth researching. You just had to take your passport to the info. office in the shop and you were issued with a card.
I ate in various places around Times Square, can't really remember any names, but do try to eat in one of the Delis, they are a New York institution and it's worth trying one of their huge hot sandwiches. (big enough for two people) There are as you can imagine lots of places to eat and you will have loads to choose from. Being on my own I didn't do much in the evening other than eat or go to a couple of shows though I did take a taxi to a Blues Club in Greenwich village on one evening.
I bought the AA essential Guide to New York it's quite small and fitted into my handbag and gave me lots of ideas and I also bought the New York Popout Map from Amazon it's also small and I found it invaluable for finding my way around.
New York is an exciting though tiring city and I hope that you and your daughter enjoy your trip as much as I did mine.