Why is the Haier 25 bottle Wine Cooler not getting cold inside?

First see if it is getting any power.

The thing has to work like a refrigerator, so here are possible causes for that:

Condenser Coils are Dirty

If the refrigerator is not cold enough the condenser coils may be dirty. The condenser is like a radiator and must stay clean in order to dissipate the heat which was removed from the inside of the refrigerator. As the coils get dirtier the refrigerator becomes less efficient, which makes it work harder to cool down. If the coils are dirty enough, the unit will never be cold enough and will not be able to cool the inside of the refrigerator to the proper temperature.

Evaporator Fan Motor

If the refrigerator is not cold enough, the evaporator fan motor might have failed. Every refrigerator has a set of coils called and evaporator. The refrigerator may have one or more evaporator fan motor(s) depending on the model and evaporator locations. The evaporator fan motor circulates the cold air from the coils thru the compartment. If there is only one evaporator it is in the freezer side. If the fan is not working, no cold air will get to the refrigerator compartment. The freezer may still get cold.

Condenser Fan Motor

If the refrigerator is not cold enough there might be something caught in the condenser fan motor blade, or the condenser fan motor might be defective. The condenser fan motor draws air over the condenser coils to cool them. The condenser fan motor is located underneath the refrigerator near the back. Pull the refrigerator out from the wall and remove the access panel to access it.

Start Relay

If the refrigerator is not cold enough, the start relay may be defective. The start relay is a small device mounted to the side of the compressor. It provides power to the run winding along with the start winding for a split second at start-up to help get the compressor started. If the start relay is defective the compressor may run intermittently or not at all and the refrigerator will not get cold enough. The start relay should be replaced if defective.

Temperature Control Thermostat

If the refrigerator does not get cold enough the temperature control thermostat might be defective. The thermostat allows power to flow through to the compressor, evaporator fan and condenser fan. If the cooling system fans and compressor are running, but the refrigerator or freezer is not cooling correctly check for an airflow or defrost system problem.

Start Capacitor

If the refrigerator isn't cold enough the compressor might be having difficulty starting. The start capacitor serves as a battery to give the compressor a little boost during start-up. If the start capacitor is burned out the compressor might not be able to start and run as often as it should. Test the start capacitor first with a capacitance meter, they don't fail often. If it's defective, replace it.

Thermistor

If the refrigerator is not cold enough the thermistor might be defective. The thermistor is a sensor which monitors the air temperature. It is connected to the control board. If the thermistor is defective the refrigerator does not cool or may cool continuously.

Temperature Control Board

If the refrigerator is not cold enough the temperature control board might be defective. The temperature control board provides the voltage to the fan motors and compressor. These boards are often misdiagnosed. Check all other components to be certain this is the cause of the problem.

Compressor

If the refrigerator is not cold enough, the compressor might be defective. The compressor is a motor which compresses the refrigerant and circulates the refrigerant through the evaporator and condenser coils. There are several other components which are more likely to be defective if the compressor doesn't work. If the compressor itself is defective a licensed professional will need to replace it.

Main Control Board

If the refrigerator is not cold enough the main control board might be defective. This is not common. Check the defrost system, cooling fans, and cooling controls first.