Upvote:3
The location you specifify is Playa d'en Bossa Beach. From the site:
Facilities: Loungers, parasols, showers, lifeguards, lockers, access for the disabled, restaurants, bars, shops
Showers are also listed on this alternate site.
Other Ibiza beaches also list showers as a facility.
Also nearby is Aguamar water park which has an entry fee of β¬18 and closes at 6pm though I doubt this will be of interest to you.
Alternatively, you may be able to visit one of the neaby hotels such as the Hard Rock Hotel as a day visitor, perhaps just for the price of a drink and take a shower at the pool.