Best value in Manhattan for 2 bed rentals?

Hi, considering relocating from London to New York and looking for a 2 bed/2 bath around 4k. Family includes myself, partner and pre-school aged child. We prefer Manhattan and will likely choose public schools. I assumed we would end up on the UES - prob Yorkville - but it looks like there is some ok value in the East Village as well. We want newish kitchen/bathroom reno type properties and no chance of vermin. Anywhere else Manhattan that anyone thinks we should consider? Fast commute (20-30 min or less) to midtown is non-negotiable. Not interested in anything above 96th St on either side. Thanks in advance for any suggestions