How to replace a monitor proprietary DC connector?
Getting the entire connector out at once with a normal iron will be very difficult.  Ideally you would use a suction tool to first remove as much solder from the holes and then try to work it out, but a high power iron is better for this.  Breaking the connector so you only have to remove one pin at a time is an option since you don't need the connector any longer, just be careful not to damage the board.  Desolder braid should also help you remove solder from the holes and clean up the pads after you remove the old part.

Another thing you could try is Chip-quik low temperature solder.  You would remove as much of the existing solder and add some of the low-temp solder.  It will stay liquid for longer so you have more time to heat all the pads, and then you should be able to remove the connector.  Once you do, be sure to remove as much as possible of the low temp solder before putting in the new connector and apply normal solder.