Further study has told me I should be using VBA and a FOR loop to re-arrange the data.
I don't know how to write such a code, but maybe if I put it in plain english, one of you can help.

FOR (each time a date appears in Column A)
--Do this on each row
--until another date appears in Column A
--Place the date in Column J
--FOR (each cell that contains a DATE)
----Take the contents of that cell and place in the next available cell in column K
----Take the contents of the cell immediately to the left of the cell and place it in column L

If I could translate this english into VBA, I would end up with three columns to the right of the initial data.
The first column would contain the dates, the second column would contain the times, and the third column would contain the temperature number associated with each timestamp.
From here it would be easy to graph temperature vs time.