Decoding the Kaadi system
JavaScript (ES6), 83 bytes
s=&gt;s.replace(s=/(^|[ṅcCjJñkKgG])[aeiou]/g,(_,c)=&gt;o=(s+s).search(c)%10+o,o='')&amp;&amp;o

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How?
We use the following regular expression to match either the beginning of the string or one of the Kaadi consonants, followed by a vowel:
/(^|[ṅcCjJñkKgG])[aeiou]/g

For each match in the input string, we invoke the following callback function that takes the content c of the capturing group as parameter:
(_, c) =&gt; o = (s + s).search(c) % 10 + o

We find the value of the consonant by looking for its position in the regular expression (coerced to a string by adding it to itself).
The consonants are ordered in such a way that their value is equal to their position modulo 10:
string   : /  (  ^  |  [  ṅ  c  C  j  J  ñ  k  K  g  G  ]  )  [  a  e  i  o  u  ]  /  g
position : 0  1  2  3  4  5  6  7  8  9  10 11 12 13 14 15 ...
modulo 10: -  -  -  -  -  5  6  7  8  9  0  1  2  3  4  -  ...

When we match the beginning of the string instead of a consonant, c is an empty string whose position in the regular expression is 0 -- which, conveniently, is the expected result in that case.
Finally, we insert this digit at the beginning of the output string o.