Translation:I work on weekdays.
It's not really clearly explained here but the Japanese language does this thing that's something like "noun-to-verb conversion". It's kinda like if you add "ing" to an English noun, you're doing that action; so add "ing" to Google, becomes googling, lower case, or using the popular search engine, which has led to the definition of "to google" as a verb (etymology-wise, "googling" came first); adding "ing" to toilet paper creates toilet papering, which in the same manner has the associated verb "to TP", or, "to cover a house in toilet paper". ~をします serves the same function, but in an increased capacity.
Having work is not the same as actually getting around to doing it... ;)
Specifically saying "do" would be contextually correct if someone had implied that you don't work on weekdays. Colloquially speaking, it's a "correction" or "determining" modifier. So, if someone said, "i never see you work!" "I do work on weekdays!" Alternately, if you specified a type of work, for instance, "i do dirty work on weekdays and wet work on weekends," this would also be correct.
へいじつ is not weekends, it is the "opposite": Every day of week that is NOT weekend.
Weekdays = へいじつ: Monday, Tuesday, Wednesday, Thursday, Friday
Weekend = しゅうまつ: Saturday, Sunday