친(親) - relatives, parents; intimate
절(切) - to cut; to slice; to disconnect
How does the second character relate to the meaning of 친절?
The character 切 has various meanings, not only 'to cut' or 'to disconnect'. According to Naver dictionary, 切 has following meanings,
a. 온통 (체)
b. 모두 (체)
When this character is used to express the meaning of a. and b., then its pronunciation is '체', not '절'. '일체' is a good example of this case.
The primary meaning of「切」means to cut.
The extended meaning of this is close, near, analogous to English cutting it very close or drawing a fine line, from which the meanings urgent and relative「親切」arise.
You don't need to memorise 10+ definitions for「切」 - the usage is mostly cut, close/near or pressing/urgent.