Here is a link to the service manual for your player.
It shows the procedures necessary to dis-assemble the player so that you can gain access to the mainboard on which the headphone jack is mounted.
It may be that the headphone jack is just loose on the mainboard (due to the headphone being plugged in and out over a long period of time) and that it can be reconnected to the mainboard. To do this you will need SMD (surface mount device) soldering expertise and tools.
If the jack is not loose but that the internal connecting springs in the jack itself have lost their tension then the jack will need to be replaced.
Here is the part number for the jack. Sony part 1-818-591-11.
Unfortunately I have not been able to find a supplier by searching online for it. You may have to contact Sony parts in your location to see if it is still available.
If repairing it yourself seems too daunting, I suggest that first see if you can get the replacement part and then contact at least 3 reputable professional electronic or computer or mobile phone repair services and ask for a quote for a repair. Advise them that the service manual is available online as this will be a help to them. I suggest contacting 3 as this way you will have a better idea of the cost if all the quotes are nearly the same.