Phonemic writing systems can certainly be useful at times. K.K. has some deficiencies but it is still by far the most commonly used system in Taiwan (although nowhere else). I forget how many sounds it has symbols for but it will be around 42.
Other systems, used internationally, are more closely or accurately based on the International Phonetic Alphabet (IPA). The IPA has a great many symbols which can represent the sounds of every language. As with K.K., the other systems for phonemic representation of English are subsets of the IPA, taking around 44 symbols from it to describe the phonemes of English.
You could learn K.K. very quickly by borrowing a chart from a student and getting him or her to run through the sounds with you. It takes a while to use it consistently, though, as there is always the temptation to transcribe a word according to how it is spelt rather than how it sounds. For example, when my fellow students and I were first practising phonemic transcription in the TESOL course, we would often add a vowel phoneme to the end of a word like ‘cake’, looking at the ‘e’ and forgetting that the final sound is /k/.
In addition, remember that K.K. and the other phonemic symbol sets are not useful for transcribing subtle differences in accent. For that, you need to use a finer transcription method and a wider range of symbols from the IPA. A phoneme is the smallest unit of sound in a language which can cause a change in meaning; thus the vowel sounds in ‘pet’ and ‘pot’ are different phonemes, but regional variations in the pronunciation of the vowel sound in ‘pet’ alone do not constitute different phonemes, because there is no change in meaning and generally the difference in sound is smaller than that between the different phonemes.
See this post and the whole thread for further information (the Karen Chung link is very interesting);
[Forumosa - Taiwan's largest and most active Taiwan-oriented global online community in English … 221#128221](Should the use of K.K. be restricted? - #10 by Joesox