a wrong thing

[azz]

a. I did a wrong thing to fix the computer.
b. I did the wrong thing to fix the computer.

c. I did a bad thing to fix the computer.
d. I did the bad thing to fix the computer.

In which case:
1. I fixed the computer but I shouldn't have. It was a wrong thing to do.

2. While fixing the computer I made a mistake and didn't fix it.

3. While fixing the computer I made a mistake but still the computer was fixed.

 

[Tdol]

They're not very natural sentences and the meaning isn't clear in any of them IMO.

1 It was wrong of me to fix the computer.
2 I tried to fix the computer [but messed up/failed]
3 I didn't fix it properly but it still worked. (This one depends very much on what you didn't do or did wrong when fixing it and why it still worked)

 

[philo2009]

a. I did a wrong thing to fix the computer.
b. I did the wrong thing to fix the computer.

c. I did a bad thing to fix the computer.
d. I did the bad thing to fix the computer.

In which case:
1. I fixed the computer but I shouldn't have. It was a wrong thing to do.

2. While fixing the computer I made a mistake and didn't fix it.

3. While fixing the computer I made a mistake but still the computer was fixed.

If you're asking about use of articles after 'wrong', either is possible: the definite article, however, will tend to be the natural choice in most cases.

The only commonly occurring case of the indefinite article that springs to mind is the set expression 'a wrong number' (meaning an incorrect telephone number).

In your example, 'the wrong thing' would be far more natural than 'a wrong thing'. However, any difficulty can be avoided by saying simply e.g. 'something wrong/incorrect'.