[Sumippi]

Hmm...simple way...?

As to 「ついたち　ふつ-か　みっ-か　よっ-か　いつ-か　むい-か　なの-か　よう-か　ここの-か　とお-か」, I think you should just accept and memorise them.

When we count something we may say 「ひと-つ、ふた-つ、みっ-つ、よっ-つ、 いつ-つ、むっ-つ、なな-つ、やっ-つ、ここの-つ、とお」. If you already know this, it may be of some help, no?

And 20th is 「はつ-か」. If you know that '20 years old' is 「はたち」, it will be easier, no?

As to the other days (from 11 to 19 and from 21 to 31), just add 'にち（nichi）' after the numbers, ex. ２４ｔｈ＝「にじゅうよん-にち」.

(oh wait--we say 'nijuu-yon-nichi' but don't say 'nijuu-shi-nichi'. And 19th is 'juu-ku-nichi', not 'juu-kyuu-nichi... !!...why?)