Её опыт работы в больнице помогает ей спокойно говорить с пациентами.

These are two different forms of она (she):

• её = her (possessive adjective, or genitive/accusative form)
• ей = to her / for her (dative form)

In the sentence:

• Её опыт работы = Her work experience (whose experience? → possessive)
• помогает ей = helps her (literally “helps to her” → dative)

In Russian, опыт (“experience”) usually takes a noun in the genitive case to say “experience of something”:

• опыт чего? работы – experience of work
• опыт преподавания – experience of teaching
• опыт жизни за границей – experience of life abroad

So опыт работы is a fixed and very common phrase meaning work experience.
Using опыт работа would be grammatically wrong.

The preposition and case depend on meaning:

• в больнице (prepositional) = in/at the hospital (location, where?)
• в больницу (accusative) = to the hospital (direction, where to?)
• на больнице is wrong in standard Russian. With больница you use в, not на, for location.

So here we need в больнице because it’s talking about where she has work experience (in a hospital), not where she is going.

The verb помогать (“to help”) takes the dative case for the person who is helped:

• помогать кому? – to help to whom?
  • помогать ей – to help her
  • помогать ему – to help him
  • помогать детям – to help the children

Using её here would be wrong, because её is genitive/accusative or a possessive, not dative.
So помогает ей = literally “helps to her” → “helps her”.

Russian uses a very common structure:

• помогать кому делать что
  = to help someone do something / to help someone to do something

So:

• помогает ей говорить = helps her (to) speak

You don’t need an extra word like “to” or чтобы here; the pattern помогать + dative + infinitive already covers “helps someone to do something”.

Also note the aspect:

• говорить is imperfective, which fits here because we’re talking about a general ability / regular behavior, not one specific conversation.

Two slightly different patterns are possible in Russian:

1. говорить с кем (with instrumental)

   • говорить с пациентами – to talk with the patients
     This implies a more two-way conversation.

2. говорить кому (with dative)

   • говорить пациентам – to speak to the patients
     This focuses more on one-way speaking (e.g., giving information).

The most natural collocation for a calm, interpersonal conversation in your sentence is говорить с пациентами (talk with patients).

Спокойно here is an adverb, formed from the adjective спокойный (“calm”):

• спокойный (adjective) – a calm person, calm voice
• спокойно (adverb) – calmly

Adverbs in Russian often end in -о and describe how an action is done:

• говорить спокойно – to speak calmly
• писать красиво – to write beautifully

So:

• спокойно говорить = to speak calmly
• спокойная would be a feminine adjective (e.g. спокойная женщина – a calm woman), not correct before a verb.

Russian word order is relatively flexible, but not all orders sound equally natural.

Your original sentence is very natural:

• Её опыт работы в больнице помогает ей спокойно говорить с пациентами.

Some possible variations:

• Опыт её работы в больнице помогает ей спокойно говорить с пациентами.
  (Slightly different emphasis: puts опыт first.)

Less natural or awkward:

• Её опыт работы в больнице помогает спокойно говорить ей с пациентами.
  (Understandable, but ей feels oddly placed.)

Don’t split помогает ей говорить too much. Keeping ей close to помогает is usually best for clarity and naturalness.

Yes, you can, with slight nuance differences:

• говорить с пациентами – to speak/talk with patients
  Neutral, very common.

• разговаривать с пациентами – to have conversations with patients
  Emphasizes the conversational, back-and-forth aspect a bit more.

• общаться с пациентами – to interact/communicate with patients
  Broader: not just talking, but overall communication and contact.

All are grammatically fine.
In many contexts, говорить с пациентами is the most neutral choice.

Её is written with ё, but in many Russian texts ё is replaced by е, so you might see ее instead.

Pronunciation:

• её is pronounced [йо] in the second syllable: [йи-йо] → ye-YO
• Stress is on the second syllable: еЁ

So:

• её опыт = ye-YO Ó-pyt (approximate English transcription)