I had a really difficult time originally writing to my chip as well. I had the NTAG216 NFC chip implanted at DEFCON this year.
The problem with my chip is that the AUTH0 field was set to 00, when it should have been set to FF. What that meant is that I had to authenticate with the default password before I could do a write. I couldn’t even change the AUTH0 field.
Eventually I found a program that allowed me to both authenticate with the default password, and then write to AUTH0 to a higher value, allowing me to write to wherever I wanted to after that.
To figure out if you had my problem, read address E3. It should be 04:00:00:FF. (Or 04:00:00:E2 if you have run the dangerous things app). Mine was 04:00:00:00.
If you need help changing the last byte of E3, let me know, and I’ll give instructions how I did it.