Either is correct.
Ability is an abstract noun. It is similar in that respect to integrity and professionalism.
We might say:
I am looking for someone with the integrity to handle cash and respect client confidentiality.
The OP has the professionalism to review his draft and to seek help when he is uncertain.
We would not say integrities or professionalisms, even though multiple instances of integrity and professionalism are listed.
Professionalism, integrity and ability are characteristics of a person. The OP has professionalism and ability. Examples of his ability are that he can communicate effectively, research, solve problems efficiently and manage his time wisely.
However it is also possible to consider each particular skill as an ability, and in that case it is not wrong to use abilities.
In the case of a covering letter the focus is on the person and the extent of his ability, so I think it is better to use "ability" rather than "abilities".