value of resistor when converting HC outputs to non-compatible inputs
The outputs of the 74HC164 and the 74HCT164 are both compatible with the AT28C256 EEPROM without any pull-up resistors.

You must have had some other error that caused incorrect operation in your previous circuit.

The 10K pull-ups should not affect operation.

The 74HC164 and the 74HCT164 are only different on the voltage levels of their inputs - the 74HCT devices are compatible with being driven by the outputs of bipolar TTL devices. TTL devices only guarantee a high output of 2.4v. This is not adequate to correctly drive a 74HC device.

74HCT devices must be powered by 5v on VCC.  74HC devices can be used for supplies of 2-5v.