Let's converge to 9!

Perl 6, 41 bytes (40 chars)

{+($_,{$_%2??[+] $_².comb!!$_/2+1}...9)}

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This uses 1-indexing of k's, so it gives 1 higher answers than the examples in OP. If this is not what the 1-indexing means, I'll have to add 1 byte more.

Explanation: It's an anonymous function. We just use the Perl 6's facility for generating lists using recursion :—). It looks like this: (first element),(block that takes the previous element and gives the next)...(end condition). In this case, the first element is $_ (argument of the main function) and the end condition is 9 (fulfilled when we generate a 9). In the middle block, we use $_ to refer to its argument ( = the previous element of the sequence). The ?? !! is the old ternary operator (better known as ? :). Finally, we take the length of this list by forcing numerical context by +(...).

The last weird thing here is the sum of digits. Numbers are Cool (behave both like strings and numbers), so we use a string method .comb on $_² (give list of characters = digits), then adding the characters up (that converts them back to numbers).

Python 2, 129 126 76 68 67 64 54 53 bytes

-3 bytes thanks to Jonathan Frech. -8 bytes thanks to Maltysen. -7 bytes thanks to Jonathan Allan. -1 byte thanks to Mr. Xcoder.

f=lambda n:n-9and-~f(n%2*sum(map(int,`n*n`))or 1+n/2)

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From somebody who probably doesn't know enough math, this seems completely arbitrary. :P

Jelly, 17 bytes

²DSµH‘$Ḃ?ßµ-n9$?‘

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Straight-forward approach. Uses 0-based indexing.

Explanation

²DSµH‘$Ḃ?ßµ-n9$?‘  Input: n
               ?   If
            n9$      n != 9
          µ        Then
        ?            If
       Ḃ               n % 2 == 1
   µ                 Then
²                      Square
 D                     Decimal digits
  S                    Sum
      $              Else
    H                  Halve
     ‘                 Increment
         ß           Call recursively
                   Else
           -         The constant -1
                ‘  Increment