That was the Whyte Block (sometimes spelt
White depending on the source), located at the corner of 107th Street. It was originally built as a mixed-use retail/apartment block in 1912. It doesn't look like it was that successful, as by 1919 it was being used to house returning veterans.
As
TimETown correctly guessed, it later became St. Joseph's Hospital. The entire building was purchased by the Catholic Archdiocese in 1927 for conversion into an "old peoples home." The Sisters of Providence were responsible for day-to-day operations. Following the construction of a new purpose built hospital next door in 1946-47 (which still stands today as
The Garneau apartments), the old building served as an annex to house the Sisters themselves. By 1962 it was in such a poor state that it was condemned. Demolition followed in 1964.
I haven't been able to confirm it (although to be fair I haven't exactly been proactively looking), but I believe it to be designed by James E. Wize, the man behind the
Alberta Hotel and
MacLean Block's design. It matches his style to a tee, particularly that of his original
Corona Hotel. If it was, it's a funny coincidence that three of his most major works (the Maclean, Corona, and Whyte) were all located on or near 107th Street, albeit on different sides of the river.
Here it is in 1914. As visible in the picture posted above, it's storefronts were completely removed when the Sisters took it over.
The Sisters' website goes into the story of St. Joseph's a bit more here:
www.providence.ca
As for the Whyte Block itself, here's a news clipping from 1962 that goes into its specific history a bit more: