WNET-DT PBS 13 is currently operating at low power on UHF 61, but will switch it's digital channel back to VHF 13 at a better power level next February after the analog shutdown. WABC-DT 7 and WPIX-DT 11 will also move from UHF to the their upper VHF channels. So as you appear to know, you will need to receive upper VHF and UHF.
If your grandmother wants to get WNET 13, you may either need a converter box with RF pass-through or put in a RF splitter to run the antenna signal to the analog tuner in the TV. But your grandmother will have to be willing to switch between the digital converter box and the TV NTSC tuner. However, a better antenna might get the weak signal for WNET-DT.
The results for the zip code show good signal levels and good line of sight for the NYC stations. Replacing the old antenna with a Winegard HD-1080 and the cable with good quality RG-6 is a good choice to try.