I am especially interested in the difference between the latter two verbs, werken and wirken.
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Arbeiten has a broad meaning. It drops in where someone or something is working or functioning.
Er arbeitet gerade an seiner Steuererklärung
Die Maschine arbeitet nicht richtig.
There also is the derivative sich abarbeiten which means to labor.
Er arbeitete sich an seiner Steuererklärung ab. (he's exhausted and probably angry.)
Werken is seldom used. It's a synonym of arbeiten then.
Lasst uns daran werken!
And then there is the school subject das Werken, which is a substantive made from that verb and means handicrafts.
Wirken in its broadest sense means to cause an effect. Use it when you want to focus on the effect of work. Most times derivatives as bewirken, mitwirken, entgegenwirken etc. are used.
Chlorix wirkt gegen Schmutz und Urinstein.
Du musst warten, bis es wirkt.
Händel wirkte hauptsächlich in Hamburg und London. (as tofro suggested)
Das bewirkt gar nichts.
Sie wirkte bei vielen wichtigen Projekten mit.
Diese Kraft wirkt der anderen Kraft entgegen.
Note the derivative verwirken has a complete different meaning: to forfeit.