لقيت المفتاح تحت السرير النهارده.

A natural Egyptian pronunciation is:

laʔeet il-miftaaḥ taḥt is-sariir innaharda

A few helpful notes:

• ق in Egyptian Arabic is often pronounced as a glottal stop ʔ, so لقيت sounds like laʔeet.
• ح in مفتاح and تحت is a strong, breathy h sound from deeper in the throat.
• الـ changes in pronunciation before some letters, so:
  • السرير is pronounced is-sariir
  • النهارده is pronounced innaharda

Here is the breakdown:

• لقيت = I found
• المفتاح = the key
• تحت = under
• السرير = the bed
• النهارده = today

So the structure is basically:

found + the key + under + the bed + today

Because لقيت already includes the subject I.

In Egyptian Arabic, verbs often show who did the action, so you do not need to say أنا unless you want emphasis.

So:

• لقيت المفتاح تحت السرير النهارده = I found the key under the bed today
• أنا لقيت المفتاح تحت السرير النهارده = also correct, but more like I found the key... with extra emphasis on I

لقيت is the past tense, first person singular form of the verb لقى / لقي in Egyptian Arabic, meaning to find.

So:

• لقيت = I found
• لقى = he found
• لقينا = we found

The -ت ending is what marks I in the past tense here.

In Egyptian Arabic, ق is very often pronounced as a glottal stop ʔ in everyday speech.

So:

• لقيت is pronounced laʔeet, not something like laqeet

This is one of the most noticeable differences between Egyptian Arabic and Modern Standard Arabic. A learner will hear this all the time in Egypt.

Because they are both definite nouns:

• المفتاح = the key
• السرير = the bed

Arabic uses الـ to mean the, just like English uses the.

If you removed الـ, the meaning would become more like:

• مفتاح = a key
• سرير = a bed

So the sentence specifically refers to a particular key and a particular bed.

This happens because of sun letters.

The article الـ is always written the same, but in pronunciation it sometimes blends into the first consonant of the next word.

For example:

• المفتاح → il-miftaaḥ
  because م is a moon letter, so the l sound stays
• السرير → is-sariir
  because س is a sun letter, so the l sound disappears and the s is doubled
• النهارده → innaharda
  because ن is also a sun letter

So the spelling stays the same, but the pronunciation changes.

Yes, النهارده can move around.

In this sentence, putting it at the end sounds very natural:

• لقيت المفتاح تحت السرير النهارده

But you could also say:

• النهارده لقيت المفتاح تحت السرير

That gives a bit more emphasis to today.

Arabic word order is often more flexible than English word order, especially with time expressions.

Yes, النهارده is a very common Egyptian Arabic word meaning today.

You may also see the spelling النهاردة. Both represent the same everyday Egyptian word.

In Modern Standard Arabic, the usual word would be:

• اليوم = today

So an MSA version would sound more like:

وجدت المفتاح تحت السرير اليوم
or
لقيت المفتاح تحت السرير اليوم in a more mixed style

But النهارده is exactly what you would expect in everyday Egyptian speech.

Yes. Depending on context, لقى / لقيت can sometimes mean met, came across, or found.

For example, a learner might hear:

• لقيت صاحبي = I met my friend
• لقيت المفتاح = I found the key

In your sentence, because the object is المفتاح and the rest of the sentence says under the bed, the meaning is clearly found, not met.

Yes. Egyptian Arabic allows some flexibility.

These are all possible, with slightly different emphasis:

• لقيت المفتاح تحت السرير النهارده
  neutral and natural
• أنا لقيت المفتاح تحت السرير النهارده
  emphasizes I
• النهارده لقيت المفتاح تحت السرير
  emphasizes today
• لقيت المفتاح النهارده تحت السرير
  possible, though many speakers may prefer the original order

The version you were given is very natural and idiomatic.

Because تحت works directly as under/beneath, and the noun after it comes normally:

• تحت السرير = under the bed
• تحت الترابيزة = under the table
• تحت الكرسي = under the chair

So this part is very straightforward:
تحت + noun = under + noun.