済む・すむ　means to end, to finish, to complete, to settle the issue, to solve the problem, etc. 例:　　あやまれば済む 　　ケガせずに済む　　金を返せば済む　　一時間ほどで済む 済まない、済みません　is the opposite of these. It means something isn't enough to end, complete, settle the issue, etc. 金を返せば済む　→　金を返すだけでは済まない・済みません。　 あやまれば済む　→　あやまったぐらいでは済まない・済みません。 So when you make an excuse or offer an apology, you say what you are doing/offering isn't enough to solve the problem, or settle the issue. 　You say ～では済みません、～だけでは済みません、etc. Hence 「済みません」has become an expression used when apologizing. (In this usage, すみません is written all in hiragana.）

You don't have to spell out すみません using 漢字。 すみません should be written すみません。 If you think all Japanse words have kanji for them, that is a misconception.There is no relation with 住む. Etymologically, however, すみません(sorry）came from the ancient and original image, ”澄む (become clear/transparent)“. 住む and 済む are derivatives of 澄む. Historically, 澄む came first, 住む and 済む next. **** What is すむ or 澄む？ Pouring muddy water in the cup, at first, the water is brown with mud, however as the time goes by, the mud goes down and the water gets transparent. This is すむ. Our ancestors applied a Chinese character ”澄” to this Japanese word すむ **** What is 住む and 済む The core image of すむ is "getting stable", which turned to mean "住む, stay in one place for a long time" and "済む：getting done, end". And also 棲む, which is 住む for animals However, they are derivatives. Again the core image of すむ is "澄む：getting transparent" The image of すみません is "I did something wrong and my heart is not quiet and peaceful like muddy water(澄まない）". . それじゃわたしの気がすみません。 However, 澄、済、住 are all Kanjis（= 中国語）, which, after all, has nothing to do with the original Japanese word "すむ". That is why you don't have to spell out すみません using kanji. BTW. はなす can be written as 話す、離す、放す, which means different but, after all, they are all the the same. It is just はなす, which means "let go / unbind"

Kanji-san, 過分のお言葉、痛み入ります。これからも、もっとがんばります！ Robertさんも、またご質問ありましたら、ぜひどうぞ。

I agree with you wholeheartedly. If you need an in-depth explanation of Japanese grammar, Aki san is the one to ask.

Thank you both for your answers. They were both very helpful and I wish I could rate them both as best answer. I have to choose one, though, so I chose AKI because he put so much effort into his answer, and he also addressed the difference between the related kanji. Again, thank you very much.