Of course, the above suggestions are assuming you actually want to resurrect the old Sony Vaio and continue using it. You said the battery is defunct. You might have to buy another one as well as a power brick. (maybe the laptop will work on the mains power with a dead battery inside?) If you think those purchases would not be cost-effective then a probably cheaper alternative would be to get the hard drive out of the laptop and put it into an external drive enclosure (a case) and then you can plug it in to another computer via USB and, (assuming the hard drive is actually working and the data is still accessible and uncorrupted) you could copy off the files, pictures, whatever and keep them somewhere else, or just use the the drive as a storage resource. If you are seeking a way to recover the items, it does sound like they are precious to you, and in this case you might wish to consider keeping copies in more than one storage location in future. (That is, keep backups).
If you are not confident of doing the above, or don't know anybody who is, most largish towns have computer repair businesses that you can take the laptop to and they will be able to supply a power brick and/or rescue the drive and put into an enclosure for you, possibly while you wait. Do a Google search and find some phone numbers. In the UK we have PC World etc but in my experience very often the smaller companies are very friendly and helpful.
