I've been to NY twice in December. The first year, there was lots of snow on the ground and it was pretty chilly. The following year, no snow but a nasty wind and it was freeeeezing!
You absolutely need a decent pair of boots. The pair I had were warm, waterproof (GoreTex) and also had decent grippy soles - important when the ground was a little slippery. All the main thoroughfares are gritted etc, however the the minor paths around the hotel etc. were not. These are the ones I had:
www.amazon.co.uk/gp/product/B003M2YCQG/ref=oh_aui_detailpage_o07_s00?ie=UTF8&psc=1
Also get decent socks to go with. I like Smartwool because of the high wool content. They are very breathable and don't smell after a day of hard use but most importantly, they're comfy (essential as you'll no doubt be walking miles each day).
I don't think your jacket will cut it to be honest, not unless you have a LOT of warm layers beneath. I had a down puffa-type jacket (from a outdoors brand, not a fashion one) and was still a bit chilly. You ideally want a VERY warm coat that is also waterproof. Or a warm coat and take a waterproof jacket with you. A wet warm coat is useless.
Tights under trousers are good. I believe that ice-skaters' tights are very warm but thin and you can buy them on Amazon. I just wore normal thick ones though. I also bought some merino wool vests.
Dont' forget decent gloves and a hat (preferably with ear flaps or get some ear-muffs!) - lots of knitted type ones let an alarming amount of wind through. The best pair of gloves I had were 99p in Tesco - microfleece ones that you buy from those hanging clip strips in the middle of the aisles! Leather would be better still.
Whatever you wear, layers are essential if you're going in and out of attractions / shops. You'll want to de-layer after 10 minutes or so of indoor heat.
Enjoy your trip. I loved NY at Christmas time and can't wait to go back one day!