Hexadecimal and the Alphabet
Jelly, 18 bytes

b⁴µ:⁵©+¹%⁵ḅ⁵ß¹®S¤?


Try it online!

The binary, 18 byte version of the source code has the xxd dump

0000000: 62 b6 8c 3a b7 85 2b 8e 25 b7 a3 b7 95 8e 88 53 83 3f b..:..+.%......S.?


and works with this version of the Jelly interpreter.

How it works

b⁴µ:⁵©+¹%⁵ḅ⁵ß¹®S¤?  Define the main link -- Left input: a (number)

b⁴                  Convert from integer to base 16.
  µ                 Start a new, monadic link.
   :⁵               Divide all base 16 digits by 10.
     ©              Save the result in a register.
      +¹            Add the quotients to the base 16 digits.
        %⁵          Take all resulting sums modulo 10.
          ḅ⁵        Convert from base 10 to integer.
              ®S¤   Take the sum of the quotients from the list in the register.
                 ?  If the result is non-zero:
            ß         Recursively call the main link.
             ¹        Else, apply the identity function.


Ḍ (decimal-to-integer) should have worked as a shorthand for ḅ⁵, but the latest version of Jelly at the time of this post had a bug that prevented me from using it.