Can't get SSD to work

I suspect you have a few issues here:

Your systems SATA port is only SATA II (3.0 Gb/s). This SSD is a dual use SSD as it can sense the systems SATA port and match it. In this case this SSD supports either SATA II (3.0 Gb/s) or SATA III (6.0 Gb/s) interfaces. It needs to be able to sense the systems SATA port correctly.

Apple had some updates in the systems firmware related to the SATA port so I would start there.

Using the HD in the system, lets start off making sure the systems firmware is up to date. Follow this Apple T/N: About EFI and SMC firmware updates for Intel-based Mac computers. Make sure your at the level listed or newer.

Next, if the EFI update didn't help you may need to check your SSD's. Sadly, you'll need a Windows system to run the Kingston SSD Manager software to check to see if the SSD's firmware is up to date. You may want to call Kingston to ask them if there is another way to check it.

After making sure all of the firmware is good, the last issue is a bad SATA cable. They are fragile! So its possible there is some slight damage that is impacting the SSD to correctly set its SATA speed so it thinks the system is able to run at SATA III which it can't.

Here is the IFIXIT guide you'll need to follow: MacBook Pro 13" Unibody Mid 2010 Hard Drive Cable Replacement, and the needed part: MacBook Pro 13" Unibody (Mid 2009/Mid 2010) Hard Drive Cable, Apple P/N 922-9062.