Why doesn't the burndown chart update to the work done in the sprint?
The burndown chart always shows the period from the start date of the sprint to the end date of the sprint. Any work done after the end date of the sprint will not be shown in the burndown chart.
So, make sure that the start and end dates of the sprint are set correctly, and cover the period where you work on the sprint. If they are not, simply change them to the correct period.
Also, weekend days are not shown in the burndown chart unless you have specified that a team in the sprint has available time on that day.
Finally, note that when you track used time, this does not itself affect the burndown chart, only changes in the "Remaining" fields do.