Stewart follows Ilyich from Austurstræti into Pósthússtræti where he enters Hótel Borg. Following a scene in the hotel restaurant between Ilyich and Slade, Stewart asks the hotel receptionist for Slade’s room number. The room number is given as 306. 306 was indeed the room in which the filming took place, now known as Suite 311. 
Hótel Borg, with its Art Deco splendour was built in 1928 for Jóhannes Jósefsson, a famous strong man and adventurer. An extract of the history of Hótel Borg is reproduced below, the full text may be viewed here:
Hotel Borg was originally built for Jóhannes Jósefsson, a famous strong-man and adventurer. Jósefsson was born in the north of Iceland in 1883 and became a champion of Icelandic wrestling at an early age. He competed in Greco-Roman wrestling at the 1908 London Olympics at the age of 25 and came in fourth. He then travelled the world, ending up in America with Barnum & Bailey’s Greatest Show on Earth, where he performed as an unarmed contender fighting opponents even armed with knives. Jósefsson became a wealthy man from his circus activities and returned to Iceland unscathed in 1927.
His dream was to build a luxury hotel in Iceland. The first millennium celebrations of the republic were at hand and a hotel, fit for foreign heads of state, was needed in Reykjavík.
Jósefsson placed great emphasis on the hotel being located in the heart of Reykjavík, which is why Hotel Borg stands dignified in a graceful square, next to the Parliament (Alþingi) and the old cathedral.
The cornerstone was laid in 1928. Eighteen months later, in January 1930, Hotel Borg opened its restaurants and four months after that, in May, the hotel itself was opened. As a proud owner of this landmark building, Jósefsson was usually referred to as ‘Jóhannes at Borg’. 
In the production stills, black and white portrait images can be seen on the walls of the hotel room. Of those identified, one is of Svein Björnsson, who was the first President of the Republic of Iceland (1944-1952). More can be read about him here.
Another is Ásgeir Ásgeirsson, who was the second President of the Republic of Iceland (1952-1968). More can be read about him here.
Images © BBC Scotland, © The Bagley Brief.
1. Gunnarsdóttir, R. (2012) pers. comm 26th November, 2012
2. Hótel Borg.is (2015). Hótel Borg History [online]. URL [Accessed Jan, 1st 2015