Welcome to BetterBidding!
Typically anywhere in midtown that has good subway access, with 'good subway' access meaning a station (or ideally multiple stations/subway lines) within walking distance. You don't need to be directly in Times Square.
Basically you would need to book 2 rooms and then call the hotel to see if they have connecting rooms and if they are willing to assign you two rooms that are connecting. Whether this is possible or not will depend upon multiple factors including the percent of connecting rooms that the hotel has, occupancy levels for your specific dates of travel, mood of the front desk clerk to be willing to put forth the effort to make the change, etc.
Just to clarify, i assume this is your maximum budget for 2 rooms per nite, correct? (so $225-$250 per room per nite).
I would tend to lean towards HOTWIRE for your situation since you need more than double occupancy in one of your rooms. All PRICELINE reservations, unless otherwise indicates, are for double occupancy, after that it's at the discretion of the hotel... whereas HOTWIRE will allow you to search for up to quad occupancy in each room. How old are your kids? (as this will give us a better idea of how much they may be willing to walk without starting to complain... 'convenient' takes on two different meanings with a 15 year old vs. a 4 year old)
I did do a quick search for your dates (searching 2 rooms/5 guests) and budget is not really going to be an issue as i'm seeing quite a few 'good' centrally located hotels that are well within budget.
Lastly, how would you rank budget, convenience, and quality of hotel... would you be willing to pay more to stay at something extra special and/or more convenient OR is budget more a priority and you would rather save money and be slightly outside of being centrally located (ie- maybe a 10-15 minute walk to a subway vs. a 5-10 minute walk to the subway)
Please remember to return to BetterBidding and use the PRICELINE and HOTWIRE links on the board to begin your travel purchases... or thru our sister site MAPSandHOTELS.com for your regular rate purchases.
Let us know the information above and we'll take it from there.