To find the correct driver you need first to identify the hardware. Click 'Start', then RIGHT click on 'Computer. From the screen that opens choose 'Device manager' by clicking on it and it will open. Run down the list of devices until you come to one that says 'Audio' and click the '+' sign at its side to expand it. You should then find the device and type for your audio, and this is what you look for on either the Acer or Realtek's web-sites to find the driver. Any driver listed against this type/model number will be correct, choose the newest.
It is always possible that it is a physical problem rather than a software one. Headphone sockets normally contain a switch activated by the jack plug to disable the speakers. It is possible for this switch to become jammed by dirt or something similar. A squirt of switch cleaner spray - if you have access to any - or a COUPLE of drops of iso-propyl alchohol into the socket followed by a few rapid insertions of the jackplug will often clear such problems. Do this with the computer switched off !
Another possibility is that, since the socket for the plug is soldered to the motherboard, the solder contacts may have failed due to repeated use. If this is the case, you will almost certainly have to do a considerable amount of dismantling and then some fairly delicate soldering. If you have doubts about your skill in these areas this really is something better left to a repair shop. If you want to find out what is involved enter 'Acer aspire 7736z-4088 teardown' into search on Youtube. If you don't find several videos on this I willl be amazed.