Just to answer the question of how to get AHT and assuming your Mac is indeed a 13inch mid-2010 MacBook Pro your model is MacBookPro7,1 and for that model you can download AHT from Apple with the link below and follow my steps below to put it on a flash drive.
Apple AHT for MacBookPro7,1 - 022-4653-A.dmg - do not use unless you've verified you indeed have a MacBookPro7,1, it won't mess up your computer if it isn't, it just won't boot.
To some these steps may not make sense, I'm just reformatting what I was told and worked when I did this for my iMac to a guide for this specific MacBook Pro, I don't dare wipe my flash drive to try another method as I don't want to loose its power. Continuing on...
(your flash drive will be WIPED)
1. Insert your flash drive of at least 16MB capacity and open Disk Utility then select your flash drive on the left and go to the Partition tab.
2. Select 1 Partition in the drop down then go to Options... and select GUID Partition Table then verify Format is set to Mac OS Extended (Journaled)
3. Finally enter a name for your drive (I recommend something simple like AHT-FD, do not use spaces) and click Apply
4. After formatting is finished mount (double click) the 022-4653-A.dmg you downloaded and copy everything to your freshly formatted AHT-FD flash drive.
5. In Finder open Go in the menu bar and select Go to Folder... then paste /Volumes/AHT-FD/System/Library/CoreServices/.diagnostics/ and click Go
6. Copy the diags.efi file and paste it to the root of your AHT-FD flash drive.
7. For the final step open Terminal and paste this code then enter your password, your computer will immediately reboot after the command completes, save any work first
cd /Volumes/AHT-FD/ && sudo bless --mount /Volumes/AHT-FD --setBoot --file diags.efi && sudo reboot
You should now be booting into AHT...
After you are finished shut down your computer, remove the flash drive, and at the start up chhime hold down Option on your keyboard and select your Mac's hard drive again.
Source for all this information: Here and here.