The correct answer is a kinase. Kinases are enzymes that couple the hydrolysis of ATP to the addition of a phosphate group to its substrate.

Phosphatase enzymes basically function oppositely to how kinases work. Phosphatases use water to hydrolyze phosphate groups off of their substrate.

Isomerase enzymes function to interconvert the structure of molecules from one isomer to another. This means that the substrate will remain with the same molecular formula, but it will have a difference in the connectivity of its bonds.

Ligases are enzymes that work by joining two molecules together.

Oxidoreductases are enzymes that act by catalyzing oxidation and reduction reactions, which involve the transfer of electrons from one molecule to another.

The last step of glycolysis, which involves the conversion of phosphoenolpyruvate into pyruvate, is catalyzed by the enzyme pyruvate kinase, yielding one pyruvate molecule and 1 ATP. The other kinases are involved in different steps of glycolysis.

Kinases catalyze the attachment of phosphate groups to their substrates. Phosphatases specifically remove phosphate groups from their substrates, which is the opposite of the function of kinases. The other enzymes listed do not have functions that involve removal of phosphate groups. 

From it's name, we can assume that this kinase will add phosphate groups to its targets. The source of these phosphate groups is ATP. PKA has 2 catalytic, and 2 regulatory subunits. When PKA is activated, there is a conformational change that causes the regulatory subunits to fall off, and frees the catalytic (kinase) portions. When PKA is inhibited, the regulatory subunits bind to the catalytic subunits and prevent phosphorylation. PKA has many activators, but cAMP is one of the most robust and well studied activator. Therefore, cAMP binds to the regulatory subunits, but does not inhibit protein function.

Polar R-groups, namely free hydroxyl groups on serine, threonine, and tyrosine, are the usual targets for phosphorylation because they are nucleophilic and can react with the phosphate group. Other R-groups are not as reactive and therefore are not ideal sites for phosphorylation. 

One of the features of the beginning of mitosis, and various steps in the cell cycle, is the increase of cyclin activity. This cyclin controls the action of Cdk, causing changes in the phosphorylation of proteins, influencing cell cycle events. Yeast cells use one Cdk, which changes cyclin throughout the cycle, while vetrebrate cells use four -- -Cdk,  /S-Cdk, S-Cdk, and M-Cdk. No matter the species, however, Cdk binds to various cyclins over the course of a cell's life.

The lysosome is the "stomach" of the cell. It contains many hydrolytic enzymes to digest and recycle the monomers used to form old polymers. Remember the opposite of dehydration/condensation synthesis is hydrolysis. Hydrolysis reactions use water to break bonds in polymers, yielding monomers that can be recycled and reused in anabolic pathways.

Lysozymes speed up by many times the hydrolysis of polysaccharides, by adding the water molecule to sugars linked in its enzyme-substrate complex. If left alone without the lysozyme, this hydrolysis would occur relatively infrequently, because it requires a large activation energy which would be supplied only by rare random collisions. The amino acid cleavage enzyme which uses the ping-pong mechanism is chymotrypsin. The enzyme which breaks nucleic acid phosophodiester bonds is phosphodiesterase. Fats are hydrolyzed by lipases, not lysozymes.

Phosphoglucomutase is responsible for altering the position of the phosphate on the glucose from the "1" position to the "6" position. However, notice how the molecular formula for the product and the substrate are the same. Enzymes that rearrange the structure of a molecule in this manner are referred to as isomerase enzymes.

Mammalian DNA ligase I has this function, and there are other DNA ligases which perform it in other animals and eukaryotes (prokaryotes also have their own DNA ligases). All the other functions mentioned are done by other classes of enzymes, not ligases (i.e. hydrolases, aminotransferases, oxidoreductases, etc.).