Well while you want to use option #2 to achieve in reality what is option #1 then depending on the current state of the vaio you have some alternatives to how you can achieve your objective which i shall go into shortly.However 1st i am curious as to why you are going the way you are when it appears that other than a power supply (source of power) issue which can be quite easily solved you are going the data recovery route.Do you intend to discard the Vaio?
Now if in fact (even though it be of short duration) if you do have some time that the vaio stays on and useable then you can transfer your data with a flash drive (least expensive).Please tell us how long currently you can use the Vaio before it depletes power from the battery.Then you can transfer the retrieved data to the HP. This may take a few transfers depending on the size of the files,the size of the flash drive,and the length of time you have access to the vaio.
The next option is to purchase a power pack that is suitable for the Vaio (a Sony one would be the most expensive). However there are units that are sold that are designed to work with your model computer that will not be made by Sony but will equally do the job and be much cheaper. They can easily be found with a google search.
Then you would be able to transfer but also able to continue the use of the Vaio with the limitation i already mentioned. Personally if all that is wrong with the Vaio is a run down battery and a lack of a power pack that is the way i would go. It should not be appreciably more expensive than my next option.
Which is to purchase a portable case that that will allow you to remove and install the Vaio HD in it and connect it to the vaio with a USB cable and run it as a slave HD to the HP.
So now you are at decision making time. We will gladly assist you to find or implement whatever route you chose to go. There are some additional details (particularly with the last suggestion that will be needed--not costing money-- that you will need to apply). But forget about just transferring the vaio HD to the HP. We remain available to assist you as required. truenorth