You could right click the relation in the relations panel and click select members. This gives you the number of ways in the selection panel. Then you could perform a search for closed ways within the selection and look at the number of ways in the selection panel again. For quicker access you could place that specific search as button in the tool bar.
See https://josm.openstreetmap.de/wiki/Help/Action/Search#Savesearchqueryinmaintoolbar and/or https://josm.openstreetmap.de/wiki/Help/Preferences/Toolbar#Previouslysavedsearchresults