Lekarka mówi, że po kilku dniach odpoczynku przestanę kaszleć, a maść i plaster nie będą już potrzebne.

Lekarka is the feminine word for female doctor.

• lekarz = doctor (traditionally the masculine form)
• lekarka = female doctor

In this sentence, the speaker is specifically referring to a woman, so lekarka is natural.

Because this is reported speech from the patient’s point of view.

The sentence means that the doctor says that I will stop coughing. So the verb has to match I, not you:

• przestanę = I will stop
• przestaniesz = you will stop

If the doctor were speaking directly to the patient, she might say something like Po kilku dniach odpoczynku przestanie pan / przestanie pani kaszleć or przestaniesz kaszleć, depending on the situation.

Że means that and introduces a subordinate clause.

So:

• Lekarka mówi = The doctor says
• że... = that...

This is very common in Polish:

• Myślę, że... = I think that...
• Wiem, że... = I know that...
• Mówi, że... = He/She says that...

Because the preposition po here means after, and after po, Polish normally uses the locative case.

So:

• kilka dni = a few days
• after po it becomes po kilku dniach

This whole phrase means after a few days.

Because kilka changes form depending on case.

Here, po requires the locative, so kilka becomes kilku:

• kilka dni = a few days
• po kilku dniach = after a few days

So kilku is just the case form required by the grammar.

Because dni odpoczynku means days of rest, and Polish often uses a second noun in the genitive to describe what kind of days they are.

So:

• dzień odpoczynku = a day of rest
• kilka dni odpoczynku = a few days of rest
• po kilku dniach odpoczynku = after a few days of rest

This is similar to English expressions like a day of rest or several days’ rest.

Because przestać means to stop, and in Polish it is commonly followed by an infinitive:

• przestać kaszleć = to stop coughing
• przestać mówić = to stop talking
• przestać płakać = to stop crying

So:

• przestanę = I will stop
• kaszleć = to cough

Together: I will stop coughing.

Because the idea is to stop an ongoing activity.

Kaszleć is the normal imperfective verb meaning to cough / to be coughing. After przestać, Polish usually uses the imperfective infinitive when talking about stopping a repeated or ongoing action.

So:

• przestać kaszleć = stop coughing

A perfective verb would suggest a single completed cough or a different nuance, which is not what is meant here.

A often links two parts of a sentence with a slight contrast, shift, or new piece of information. It can mean something like and, while, or and as for depending on context.

Here:

• przestanę kaszleć = I will stop coughing
• a maść i plaster nie będą już potrzebne = and the ointment and plaster will no longer be needed

Using a sounds very natural because the sentence moves to a second related result. I would simply join items more neutrally, while a gives a smoother contrast/transition.

Because maść i plaster is a compound subject: two things are being talked about together.

So the verb must be plural:

• będą = they will be

And the adjective must also be plural:

• potrzebne = needed

If there were only one item, it would be singular:

• Maść nie będzie potrzebna.
• Plaster nie będzie potrzebny.

Because maść and plaster are things, not people.

In Polish plural adjective forms distinguish:

• masculine personal: potrzebni — used for groups including men/people
• non-masculine-personal: potrzebne — used for things, animals, and many non-person groups

Since maść and plaster are objects, the correct form is potrzebne.

In a negative sentence like this, już usually means any longer or any more.

So:

• będą potrzebne = they will be needed
• nie będą już potrzebne = they will no longer be needed / they won’t be needed any more

This is a very common use of już.

Right — here plaster is a medical plaster, usually an adhesive bandage / Band-Aid / patch, not building plaster.

In medical contexts:

• plaster = plaster, bandage patch, adhesive dressing
• maść = ointment

So maść i plaster is a very natural medical combination.

Because Polish punctuation normally separates clauses more clearly than English.

• A comma before że is standard when it introduces a subordinate clause.
• A comma before a is also standard when it joins two clauses.

So:

• Lekarka mówi, że...
• ..., a maść i plaster...

These commas are required in normal written Polish.