Gauni la Asha lina lebo ndogo yenye jina lake ndani.

Because the head noun gauni is in noun class 5. The “of” connector (associative) agrees with the head noun:

• Class 5 (gauni): la → gauni la Asha (Asha’s dress)
• Class 6 plural (magauni): ya → magauni ya Asha (Asha’s dresses)

If the head noun were class 9, you’d use ya (singular) or za (plural), e.g. lebo ya gauni, lebo za magauni.

The verb “to have” is built as “subject marker + -na.” The subject marker must match the class of the subject.

• gauni (class 5 singular) → li- + -na = lina → Gauni … lina …
• lebo (class 9 singular) → i- + -na = ina → Lebo … ina …

So with gauni, you need lina.

Yenye means “which has/with,” and it agrees with the noun it describes. Here it describes lebo (class 9), so the form is yenye:

• lebo yenye jina lake = a label that has her name

Some useful pairs:

• lebo (cl.9 sg) → yenye: lebo yenye…
• lebo (cl.10 pl) → zenye: lebo zenye…
• gauni (cl.5 sg) → lenye: gauni lenye…
• magauni (cl.6 pl) → yenye: magauni yenye…

Yes. yenye is a compact way to say “that has/with.” The longer relative construction is also correct:

• lebo ndogo iliyo na jina lake = a small label that has her name

Possessive adjectives also agree with the noun possessed. Jina is class 5, so “his/her” is lake:

• class 5: langu, lako, lake, letu, lenu, lao → jina lake
• class 9/10: yangu, yako, yake, yetu, yenu, yao → lebo yake

So you say jina lake, but lebo yake.

As written, ndani (“inside”) is somewhat ambiguous at the very end. Readers will usually understand it to mean the label is inside the dress. To be crystal clear:

• Location of the label: … lebo ndogo iliyo ndani ya gauni yenye jina lake.
• The label has her name on it (and is inside the dress): … lebo ndogo yenye jina lake ndani ya gauni.

Use:

• ndani alone as an adverb (“inside” in general): … ndani.
• ndani ya + noun to specify “inside of X”: ndani ya gauni (inside the dress).

So both are fine; ndani ya gauni is just more specific.

Adjectives follow the noun they modify. Hence:

• lebo ndogo (a small label)
• gauni jipya (a new dress) You generally don’t put the adjective before the noun.

Yes:

• gauni lake = her dress (class-5 possessive)
• gauni la Asha = Asha’s dress

Both are correct. gauni lake la Asha is redundant and unnatural; pick one.