It depends on whether the text values *CAN* be converted to DATETIME.
I would play it safe and do it in steps, if I were you.
Look at the results. *ARE* those textual dates in the correct order? If so, then obviously the format of the textual dates *can* be converted by MySQL and you are nearly home free.
SELECT c_last_edit FROM yourtable ORDER BY CAST(c_last_edit AS DATETIME) LIMIT 40
*IF SO*, then I would STILL do the "fixup" in at least two more stages:
*IF THE FIRST TEST FAILS*
(1) ALTER TABLE yourtable ADD c_last_edit_dt DATETIME;
(2) UPDATE TABLE SET c_last_edit_dt = CAST(c_last_edit AS DATETIME)
(3) [optional] remove the c_last_edit field and from now on use c_last_edit_dt
That is, if there are values in that text field that can't be just CAST to DATETIME, then you have more work to do.
In that case, can you show us 20 or so sample values from that field, as they now are in your DB table?