At pH 12, all the acidic hydrogens will have been removed.
We usually write the structure of lysine as
In very acidic solutions (pH 1), both NH₂ groups will be protonated, as in structure A below.
COOH = 2.2
NH₂ = 9.0
NH₂ in side chain = 10.5
As you make the solution more basic, you will remove the acidic protons one after the other.
By the time you reach pH 12, you will have removed all of the acidic protons.
Lysine will exist as structure D.