Learn Hardcore Japanese: kanozyo to nakanaorisite totemo uresikatta desu.

If the context is about a romantic relationship (for example, they had a fight as a couple), 彼女 usually means girlfriend.
If it is just talking about a female person and there is no romantic implication, it can mean she or her.

In a sentence like 彼女と仲直りしてとてもうれしかったです, many learners (and many native speakers) will naturally read 彼女 as my girlfriend, unless the prior context clearly says she is, for example, a classmate, coworker, or sister.

Why is the particle と used after 彼女? What does it do here?

In this sentence, と after 彼女 means with.

彼女と仲直りする = to make up with her / to reconcile with her

This と is the same と you see in other expressions of doing something together or with someone:

友だちと遊ぶ – to play with a friend
先生と話す – to talk with the teacher

So 彼女と is with her, not and she in this context.

What exactly does 仲直りして mean? Is 仲直り a verb?

仲直り by itself is a noun meaning reconciliation or making up (after a fight).

When you add する, it becomes a suru-verb:

仲直りする – to make up, to reconcile

In the sentence, you see 仲直りして, which is the te-form of 仲直りする:

仲直りする → 仲直りして

So structurally, it is:

彼女と仲直りして
with her + make up (te-form)

Meaning: having made up with her / after making up with her / making up with her and …

Why is 仲直りして in the te-form? What is it doing before とてもうれしかったです?

The te-form in 仲直りして links this action to the following feeling うれしかった.

Here the te-form is mainly showing a reason or cause:

彼女と仲直りして、とてもうれしかったです。
→ I was very happy (because I) made up with her.

You can think of it as:

彼女と仲直りしたので、とてもうれしかったです。
(I made up with her, so I was very happy.)

So 仲直りして does not mean making up (now); instead it connects the fact that we made up to the resulting emotion I was happy.

Why is 仲直りして not in the past tense, but うれしかった is? Isn’t that a tense mismatch?

It looks like a mismatch, but it isn’t.

仲直りする → 仲直りして (te-form)
うれしい → うれしかった (past)

The te-form (仲直りして) has no tense by itself; its tense is understood from the main verb or main predicate that follows it. Here, the main predicate is うれしかった (past), so the whole situation is understood as past:

[having made up with her] → I was very happy.

If you wanted a clearly present-time sentence, you would say something like:

彼女と仲直りして、とてもうれしいです。
I am very happy (that I have made up with her).

So the time reference comes from うれしかった, and 仲直りして shares that past time.

Where is I in this sentence? Why is the subject missing?

Japanese often omits the subject when it is clear from context.

In English we must say I was very happy, but in Japanese:

(私は) 彼女と仲直りして、とてもうれしかったです。

The 私は is simply left out because the speaker is talking about their own feelings, which is usually obvious in conversation.

So the implied full sentence is:

私は彼女と仲直りして、とてもうれしかったです。
I was very happy to make up with her.

What is the role of とても? Could it go somewhere else in the sentence?

とても is an adverb meaning roughly very.

Here it modifies うれしかった:

とてもうれしかったです。 – I was very happy.

Common placements:

彼女と仲直りして、とてもうれしかったです。
彼女と仲直りして、うれしかったです。とても。 (more colloquial, spoken, with a pause)

The most natural for learners is exactly as in the original sentence: just before the adjective it modifies.

You will also hear casual variants with a similar meaning:

すごくうれしかった
とってもうれしかった

These are less formal than とても.

Why is it うれしかったです and not just うれしかった? Can an adjective be followed by です?

Yes, adjectives in the past tense can be followed by です to make the sentence more polite.

Base forms:

うれしい – (I am) happy
うれしかった – (I was) happy

Polite forms:

うれしいです – I am happy (polite)
うれしかったです – I was happy (polite)

Both are grammatically fine:

うれしかった。 – plain, casual
うれしかったです。 – polite, suitable in most everyday polite situations

So うれしかったです is simply the polite-past version of うれしい. It is not wrong or redundant.

Is うれしかった a verb or an adjective? It looks like a past-tense verb.

Grammatically, うれしい is an i-adjective, and うれしかった is its past form, not a verb.

Japanese i-adjectives can directly show tense, which is different from English:

うれしい – happy (non-past)
うれしかった – was happy (past)
さむい – cold (now / in general)
さむかった – was cold

So うれしかった behaves a bit like a verb meaning was happy, but in Japanese grammar terms it is still an adjective in its past inflection.

Are there other natural ways to say this, and do they change the nuance?

Yes, there are a few common variations, each with a small nuance difference:

彼女と仲直りできて、とてもうれしかったです。
- Adds できて (was able to), emphasizing that you are glad you were able to reconcile.
彼女と仲直りして、本当にうれしかったです。
- 本当に (truly, really) gives a slightly stronger, more heartfelt feeling than とても.
彼女と仲直りして、すごくうれしかった。
- Casual, すごく instead of とても, and no です. This sounds like talking to a close friend.

All of them keep the same basic structure:
[彼女と仲直りして] + [degree adverb] + うれしかった (です) – I was (very/really) happy to make up with her.