Hun har svært ved at læse synstavlen og kniber øjnene sammen.

Har svært ved at is a very common Danish expression meaning has difficulty ...-ing or finds it hard to ....

Structure:

• have
  • svært ved
    • at
      • infinitive

So:

• Hun har svært ved at læse = She has difficulty reading / She finds it hard to read

You can use it with many verbs:

• Jeg har svært ved at sove. = I have trouble sleeping.
• De har svært ved at forstå det. = They have difficulty understanding it.

Because svært ved at is a fixed construction. After ved, Danish normally uses at + infinitive in this expression.

So you say:

• Hun har svært ved at læse.

Not:

• Hun har svært ved læse.

Even though ved is also a preposition in other contexts, here the whole phrase have svært ved at ... functions as one standard pattern.

Because after at, Danish uses the infinitive form of the verb.

• at læse = to read
• læser = reads / is reading

In this sentence, the main finite verb is har:

• Hun har svært ...

Then at læse is the infinitive phrase connected to it:

• Hun har svært ved at læse ...

Synstavlen means the eye chart.

It is a compound noun:

• syn = sight / vision
• tavle = board, chart, tablet-like display
• synstavle = eye chart
• synstavlen = the eye chart

Danish often makes compound nouns this way, and then adds the definite ending to the whole word.

Because Danish usually marks the by adding an ending to the noun.

For a common-gender noun like synstavle, the definite singular is typically -en:

• en synstavle = an eye chart
• synstavlen = the eye chart

This is different from English, which uses a separate word (the).

It means squints or more literally squeezes her eyes together.

• knibe = to pinch, squeeze, narrow
• øjnene = the eyes
• sammen = together

As a full expression, knibe øjnene sammen means to narrow or squeeze the eyes, usually because you are trying to see better, are reacting to light, or are showing strain.

Because sammen is part of the verbal expression knibe ... sammen, and in a main clause it often appears later in the sentence.

Danish often splits these verb-particle expressions:

• Hun kniber øjnene sammen.

Compare English phrasal verbs:

• She shuts her eyes tight
• She squeezes her eyes shut

So kniber and sammen belong together in meaning, even though another word can come between them.

Because Danish often uses the definite form for body parts where English might use a possessive.

So Danish says:

• Hun kniber øjnene sammen.

Literally: She squeezes the eyes together

But natural English says:

• She squints
• She squeezes her eyes together

This pattern is common in Danish:

• Han vasker hænderne. = He washes his hands.
• Jeg lukkede øjnene. = I closed my eyes.

Because the subject is the same for both verbs, so Danish can leave it out in the second part.

• Hun har svært ved at læse synstavlen og kniber øjnene sammen.

This means:

• She has difficulty reading the eye chart and squints.

You could repeat hun, but it would usually sound unnecessary here:

• Hun har svært ved at læse synstavlen, og hun kniber øjnene sammen.

That version is also correct, just slightly more explicit.

Because the sentence describes what is happening in a general present-time situation.

• har = has
• kniber = squints / narrows

Danish often uses the present tense where English might also use a progressive form:

• Hun kniber øjnene sammen. = She squints her eyes / She is squinting

So the Danish present tense can cover both simple present and present progressive meanings depending on context.

Yes, that is a helpful way to think about it.

The core meaning comes from knibe ... sammen as a unit:

• knibe = squeeze/pinch
• sammen = together

In actual sentences, the object often goes between them:

• Hun kniber øjnene sammen.

This is similar to how English phrasal verbs can be split:

• turn the light off
• pick the book up

So sammen is not random; it is part of the verbal expression.

Sometimes yes, but the meaning changes slightly.

• læse synstavlen = read the eye chart
  This focuses on identifying the letters or symbols on it.
• se synstavlen = see the eye chart
  This focuses more on whether she can visually make it out at all.

In an eye-test context, læse synstavlen is very natural because the person is usually trying to read letters from the chart.

Yes, it is natural.

A few alternatives are:

• Hun har svært ved at læse synstavlen og myser.
  = She has trouble reading the eye chart and squints.

• Hun kan ikke læse synstavlen ordentligt og kniber øjnene sammen.
  = She can’t read the eye chart properly and squints.

• Hun har svært ved at se, hvad der står på synstavlen.
  = She has difficulty seeing what is written on the eye chart.

Your original sentence is perfectly idiomatic and clear.