You need to look at NYC public schools.
The chances of getting into a private school for the second semester are slim to none, but you could try RC schools. A child of five isn't going to suffer too badly from basically reinforcing learning already done in Reception. Don't dismiss the American approach to K-3 education. It's not deficient, just different.
I think your child will be lonely if she doesn't go to school, and you and your H will also miss out on the company of other parents and a big chunk of American/ NYC culture, which is half the fun of spending time there imo. There will be birthday parties, playdates, etc for your child - a lot more fun than life without company. There's only so much library and museum visiting a child of five will appreciate.
Assuming the other language is French?
frenchlanguagek12.org/4151-french-dual-language-programs-nyc-public-schools
If your H's company is providing an apartment, do they know you will need to keep school for DD in mind?
Double check where the company's apartments are located. Look at catchments.
I assume if a company apartment it will be furnished and with all necessary appliances.
I suspect you won't be able to work for a six month secondment, but you might be able to work remotely for whatever employer you currently work for. DH's company needs to look into tax and visa implications for you, or your employer can figure it out.
Your child will need to be up to date with her vaccinations in order to go to school, either public or private. You can find out what NYC requires online here:
www.schools.nyc.gov/docs/default-source/default-document-library/health-and-wellness/immunization-requirements-letter-and-chart
KIN wear is a line of satin lined hoodies. They're really nice. I have one.
kinapparel.org/