Since m_elapsed is a variable, it needs to be outside the quotes when building m_strSQL, i.e.
Code:
m_strSql = "update AppDataFile set Lang_Time = '" & m_elapsed.value & "' where AppId =" & CStr(Form_Examination.AppID.Value) & ";"
I find the best way to create these type of SQL statements is to manually create an example of a query that does what I want. Then switch to SQL view and copy/paste the code that results. This is pretty much exactly what the code you are building needs to look like.
Then, in your VBA code, temporarily comment out the line that runs your SQL code (the "Execute" line), and replace it with a MsgBox that returns the SQL code you just built, i.e.
Then compare it to the code you copied out earlier. Does it look similar? Does it have the correct values, quotes, etc in all the right places?
Once you are satisfied that it does, you can uncomment the VBA code that runs the SQL code, and get rid of the MsgBox.