How do I remove the wiper switch on a2001 discovery 2

if you are careful the job is not too hard but you may need to remove the steering wheel with a gear puller to get at the screws which hold the switch in place.

You have a blown fuse or a bad wiper motor.

there is a circuit borad in the wiper motor that may have went out it is not very expensive to fix

it may be that the wiper arm is worn on the inside of the bottom bushing you will need to replace it, make sure you turn off the wipers before you turn off ignition so the wipers travel to their proper rest spot on windows some just lift off some have a screw in them just replace them in the same rest position as for any special tools you will need to contact a good garage to see if any are required or your local Land Rover Dealer

Likely the relay is the problem, the multifunction switch can fail as well.
I have the troubleshooting diagram for electrical on Series 1 Discos but a very good working knowledge of electrical circuitry is necessary to use this info.

You meant the wiper do not park? That is a problem with the motor assembly, which is responsible for parking the wipers. The switch just turns it on, and off.

It all depends on the tools you have and if you have patience....this is the task for removing and replacing the switch:

Removal & Installataion
CAUTION These models covered are equipped with a Supplemental Restraint System (SRS), which uses an air bag. Whenever working near any of the SRS components, such as the impact sensors, the air bag module, steering column and instrument panel, disable the SRS and refer to the precautions in the Chassis Electrical Section.

1. Make sure the steering wheel is in the center position (wheels in the straight ahead position)
   
2. Remove the key from the ignition.
   
3. Disconnect the negative battery cable first, then the positive cable.
   
4. Wait 10-20 minutes for the back-up power to discharge. Do NOT proceed until the back-up power is discharged from the SRS system!
   
5. Remove the driver’s side air bag module and steering wheel, as outlined under the Steering Wheel Removal & Installation procedure in this section.
   
6. Remove the steering column nacelle, as follows:
   
   • Open the fascia lower access panel.
     
   • Remove 3 screws securing steering column nacelle halves together.
     
   • Release clips securing nacelle together and remove from steering column.
     

Exploded view of the steering column nacelle (cover) mounting

Remove the rotary coupler, as follows:

• Release rotary coupler connector from bracket and disconnect from harness.
  
• Disconnect rotary coupler connector from harness.
  
• Remove rotary coupler.
  

View of the rotary coupler

1. Release passive coil from steering column lock and lower the coil without disconnecting the harness.
   
2. Release front wiper delay connector from column switch mounting bracket and disconnect from harness.
   
3. Disconnect harness connector from wiper switch.
   
4. Remove 2 screws securing switch, carefully raise edge of switch bracket to clear location peg and remove switch.
   

Release front wiper delay connector (3) from column switch mounting bracket and disconnect from harness. Disconnect the wiper switch connector (4), then remove the switch mounting screws (5)

To install:

1. Install the new switch to bracket and secure with screws.
   
2. Connect harness connector to switch.
   
3. Connect front wiper delay connector and Install the connector to switch bracket.
   
4. Install the passive coil to column lock.
   
5. Install the rotary coupler, as follows:
   
   • Install the rotary coupler to steering column and connect connector to harness.
     
   • Connect rotary coupler connector to harness and secure to bracket.
     
6. Install the steering column nacelle, as follows:
   
   • Position the nacelle to steering column and secure with clips and screws.
     
   • Close fascia lower access panel.
     
7. Install the steering wheel and driver’s air bag module, as outlined in this section.
   
8. Connect the positive, then the negative battery cables.
   

check to see if your wipers work, if so then it's not a switch. Does the rear spray and wiper work also. It could be a bad washer pump located in the washer bottle, or a clogged line to your front wipers.

It sounds lilke the wipers are just going through the "park routine": one swipe and stop.

The melted part is the end of a wiring harness. But replacing this will not solve the problem. It sounds like you have a bad wiper motor which is pulling too much current and overheating.

There are switches on the wiper motor which probably have burned out and the only one left working is the park switch.

This may be your wiper switch on your steering column very common problem.