Hi,
It's 3.55am over here and I should be asleep, so I might be on the wrong track entirely..
I don't think you need a new hhandle or button. I'm guessing when it was freezing in the past it put a strain on the actuator (electronic solenoid switch) which unlocks that door. That is not an easy DIY fix but there are instruction on here somewhere I think or you could try youtube... most Hyundai/Kia model would be very similar as in how they work/look internally I imagine.
As far as sourcing a replacement goes, a secondhand part from a wreckers would be a cheaper but riskier option than a new part from a dealer or online store. With the current situation sourcing parts can be a little harder/slower unfortunately.
Hope that helps a bit. Others mighy have parts numbers and/or different opinions.