In choosing your enclosure in which to mount the old internal HDD, you need to look for three details:
1. Interface between HDD unit and enclosure. What is the HDD type - IDE or SATA - of the old drive from the HP laptop? Whichever it is, that is what you need inside the enclosure to plug in the drive. "Interface" here means the electrical AND mechanical layout of the connectors in the enclosure must match the connectors on the HDD so it can just plug in. Now in your case the old drive came from an HP laptop, and laptop IDE drives certainly are not the same as desktop IDE drives. I am not sure whether the connections on laptop SATA drives are the same as desktop SATA drives. So be sure whether the enclosure is really suited to the LAPTOP version of the drive you have. This MAY also impact the physical layout for how the HDD is mounted securely inside the enclosure.
2. The interface from enclosure to computer. For this you need to know what ports / connectors your new Toshiba unit has available. Almost all machines have USB2 ports, some have USB3. Some have eSATA, and a few have IEEE 1394a (aka Firewire 400). Of these, USB2 is slower than the others, but is almost universal. So pick an enclosure that has a port matching what the Toshiba has available. Some enclosures will offer a combination of two (sometimes three) external interfaces in one case, to be used one at a time.
3. Power. I prefer that power to the HDD in the enclosure be supplied by a separate power supply "brick" that comes with the enclosure; that way there is none drawn from the computer. However, there are many external drives, and enclosures to make them, designed specifically for laptop machines that try to adhere to the small-and-simple design concept. So there are lots that use small low-power HDD's inside and then use the USB2 port to supply all the power it needs, with NO extra power supply. In some cases the power available on a USB port (which is limited) is not quite enough, and the external drive unit comes with a cable that has TWO USB connectors on the end. Both must be plugged in to get enough power to run the drive. So you can decide what sort of arrangement appeals to you for enclosing a small portable HDD.