This is a unit of distance, even though it may not look like it.
It is the distance a person can walk in fifteen minutes.
The hotel is at X. The campus is at Y. The distance between X and Y is one fifteen-minute walk.
All distances are expressed in a number of distance units.
one mile, two miles, three miles, ...; one foot, two feet, three feet ...;
one two-hour drive, two two-hour drives, three two-hour drives, ...
one fifteen-minute walk, two fifteen-minute walks, ...
(Of course, we combine hours or minutes so two two-hour drives is really one four-hour drive.)
We can say
a in place of
one in all these cases.
a means
one.
a mile = one mile; a foot = one foot; a two-hour drive = one two-hour drive; a fifteen-minute walk = one fifteen-minute walk.
the is not used for counting in this way.
the cannot represent a number, so
the cannot be used here.
CJ