Transferring Mail using Target Disc Mode

Please I am begging for someone to help me ! I have an older Macbook Pro , that is running , or was running Mavericks 10.9.2. Awhile ago it quit booting and a reinstal using recovery was unsuccessful. Using disk utility i was able to determine the drive is fine and it is in fact an OS X issue. It was time to buy a new computer anyways , so I did. Now connecting the two via fireware using Target Disc Mode I am able to find and transfer all that I have wanted , except for my previous mail folders.

when i look through the folders on the drive of the computer that wont boot , there are two library folders listed. One is all on its own , and one is a sub folder inside the system folder. The only mail folder i can find in either of those folders only carrys previous stationary information. I care not about the stationary! There is no library folder listed inside of my user folder , which makes me wonder , can i access my user library on a dead macbook using target disc mode. Or can i not get access to it this way.

any help is appreciated!