Medica sperat aquam tepidam et quietem rigorem quoque lenire posse.

Medica is nominative singular feminine, so it is the subject of sperat.

Here it is best understood as a noun: female doctor or woman physician.
It comes from the same adjective family as medicus / medica / medicum (medical, healing), but in context it is being used substantively, as a noun.

So:

• medica = the female doctor
• sperat = hopes

Together: The doctor hopes ...

Because Latin often expresses this idea with an accusative-and-infinitive construction instead of a clause with that.

In English, we say:

• The doctor hopes that warm water and rest can relieve stiffness.

In Latin, this becomes more literally:

• The doctor hopes warm water and rest to be able to relieve stiffness.

So Latin uses:

• sperat = hopes
• then an infinitive phrase: aquam tepidam et quietem ... lenire posse

This is a very common Latin pattern after verbs of thinking, saying, knowing, perceiving, and often hoping.

They are in the accusative because they are the subject of the infinitive in the accusative-and-infinitive construction.

That can feel strange to an English speaker, because in English the subject would stay a normal subject:

• warm water and rest can relieve stiffness

But in Latin, inside indirect statement:

• aquam tepidam et quietem ... posse

So:

• aquam tepidam = warm water
• quietem = rest

These are not the direct object of sperat. They are the things that are hoped to be able to relieve.

Rigorem is also accusative, but for a different reason: it is the direct object of lenire.

So there are two different uses of the accusative in the sentence:

1. aquam tepidam et quietem
   accusative because they are the subject of the infinitive

2. rigorem
   accusative because lenire takes a direct object: to soothe / relieve something

So the structure is:

• Medica sperat
  the doctor hopes

• [aquam tepidam et quietem]
  warm water and rest

• [rigorem]
  stiffness

• [lenire posse]
  to be able to relieve

Because posse means to be able, and it is normally followed by another infinitive.

So:

• lenire = to soothe, relieve, ease
• posse = to be able

Together:

• lenire posse = to be able to relieve

This is just like English:

• can relieve
• or more literally, to be able to relieve

Latin uses the infinitive posse, and posse itself takes the infinitive lenire.

Because posse is an infinitive, and infinitives do not change for person or number.

In English, we might think:

• water and rest can ...
• so maybe we expect something like can for plural

But Latin is not using a finite verb there. It is using the infinitive posse inside indirect statement.

So the idea is not:

• they can

but rather:

• to be able

Since infinitives do not show singular/plural, posse stays the same no matter whether the accusative subject is singular or compound.

Tepidam agrees with aquam.

Both are:

• feminine
• singular
• accusative

So:

• aquam = water
• tepidam = warm/lukewarm

Latin adjectives agree with the nouns they describe in gender, number, and case.

That is why it is:

• aquam tepidam

and not aquam tepidum or some other form.

Because its form shows that it agrees only with aquam.

The sentence says:

• aquam tepidam et quietem

That means:

• warm water and rest

not:

• warm water and warm rest

If Latin wanted the adjective to describe both nouns, the wording would need to be different, and in any case quietem would need its own matching form if the writer wanted to say warm rest.

So here the adjective belongs only to aquam.

Quies is the nominative form of the noun meaning rest.
Here the sentence needs the accusative form, because quietem is part of the accusative-and-infinitive construction.

So:

• nominative: quies
• accusative: quietem

This is a third-declension noun, so the forms change more noticeably than in first-declension nouns like aqua.

Quoque means also or too.

It often comes after the word it especially emphasizes. So:

• rigorem quoque

most naturally means:

• stiffness too
• stiffness also

This suggests that warm water and rest can relieve stiffness as well as something else already mentioned or understood from context.

English usually puts also before the word or later in the clause, but Latin commonly places quoque after the emphasized word.

The word order is perfectly good Latin, even though it is less rigid than English.

A helpful way to read it is by grouping the parts:

• Medica sperat
  the doctor hopes

• aquam tepidam et quietem
  warm water and rest

• rigorem quoque
  stiffness too / also stiffness

• lenire posse
  to be able to relieve

Latin word order is flexible because endings do much of the grammatical work. So instead of relying mainly on position, you identify the function of each word by its form.

Lenire means to soften, to soothe, to ease, or to relieve.

With rigorem, it means something like:

• to relieve stiffness
• to ease rigidity
• possibly to lessen chills/rigor, depending on context

So the exact English wording can vary a little, but the core idea is making rigor/stiffness less severe.