Imparfait ou passé composé?

J.S. 

This is a piece of background information, a reason for the ‘ being late ‘ , which is the event described. The speaker is not describing a notable ‘ event ‘ specifically in this phrase.

The ‘ washing up ‘ itself may have been an event, but as it follows the expression “  il faut que “, the required verb in that clause is in the present subjunctive - ‘ lave ‘.  

However the ‘ needing/ obligation to do (so) ‘ which is being mentioned here, is relatively indefinite. 

On both counts, imparfait is the tense used in everyday French hence ‘ il fallait que ‘ . 

The links below may help with understanding the usage of imparfait to ‘ describe something ‘ that is background to the action.

 Expressing opinions and describing with the imperfect tense in French (L'Imparfait)

 https://www.frenchtoday.com/blog/french-verb-conjugation/passe-compose-versus-imparfait/