Most NY source remote income is taxable in NY. Although you did not live there, you are considered as working in NY if you worked remotely from home. In your case, you would file a NY nonresident return on all of your income, and Illinois will give you credit for the NY income that you earned while working remotely from IL, since each taxes the amount of income you earned in NY during the same time period, so you pay NY and get the credit on the IL return.
The income will impact your quarterly taxes so you should estimate the tax liability for NY and for the IRS and make the according payments. You can use TurboTax tax preparation software to do the calculations for you, or get a copy of the worksheet accompanying Form 1040-ES and work your way through it.