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The conjugate acid of aspartic acid is [COO-] (the carboxyl group). The pKa that you will use in the Henderson-Hasselbalch equation is the pKa of the carboxyl group, which is 2.88. The concentrations of the amino acid and its conjugate acid are not given in the image, so you cannot calculate the pH of the solution using the Henderson-Hasselbalch equation.
However, you can use the pKa of the carboxyl group to estimate the pH range at which aspartic acid will be mostly protonated (i.e., exist as the zwitterion) and mostly deprotonated (i.e., exist as the anion). The Henderson-Hasselbalch equation tells us that the pH of a solution is equal to the pKa of the acid when the concentrations of the acid and its conjugate base are equal. This means that at pH 2.88, half of the aspartic acid molecules in solution will be protonated and the other half will be deprotonated.
At pH values below 2.88, the majority of aspartic acid molecules will be protonated. At pH values above 2.88, the majority of aspartic acid molecules will be deprotonated.
Therefore, the pH range at which aspartic acid will be mostly protonated is pH 2.33-3.46. The pH range at which aspartic acid will be mostly deprotonated is pH 3.46-4.07.