Working out AT28C16 LED output from datasheet
Normally Voltage Output High will be a little less than VCC, as most ICs do not use internal boost circuits that could make it possible to be higher.

In this case, you could look at the Absolute Maximum rating:

Absolute Maximum Ratings*
Temperature Under Bias............... -55°C to +125°C
Storage Temperature ................. -65°C to +150°C
All Input Voltages (including NC Pins)
with Respect to Ground ...............-0.6V to +6.25V
All Output Voltages
with Respect to Ground ...............-0.6V to VCC + 0.6V
Voltage on OE and A9
with Respect to Ground ...............-0.6V to +13.5V


So the max you should ever see on this IC is VOH as (VCC - GND) + 0.6V. But it would likely be 5V.

So using Ohm's law for the led resistor, (5V - 2.2V) / 330 = 0.008A or 8 mA. But since the LED would not show 2.2 volts forward drop at 8 milliamps, then you should adjust the resistor up a bit. 470Ω or 500Ω may be better.

In any case, you are using the EEPROM in a non-standard way, which means it does not have specs for this type of operation. There is no way to know how well the outputs will deal with 8mA on each of the pins, if the voltage drops due to the higher current, if they will blow. The VOH spec is only tested at 0.4mA source.