The Fitness Center at Hotel Holt - The Art Hotel
Hotel Holt's fitness center is mentioned by experts. Out of 5 professionals tracked on TripExpert, 2 endorsed the fitness center, including Fodor's and Star Service.
Hotel Holt - The Art Hotel 
 Fodor's
 Star Service
 "There is a fitness room in the hotel and all guests get free access to the largest gym and swimming pool in Iceland (a seven-minute cab ride away)" 
 "The small fitness center offers modest equipment, and guests have free admission to the best fitness center in Reykjavik"