What conjugation of 送る is used in 返事をしたつもりだったが、送れていなかった

This 送れて is the te-form of the "potential form" 送れる of 送る.

The potential form is sometimes used to mean that something was done successfully

メールを送れた
I (successfully) sent the email (lit. I was able to send the email)

In the sentence

返事をしたつもりだったが、送れていなかった

the te-form (which together with いる may be used to express a state) is used to express a state of "not having been sent (successfully)", so

I was under the impression that I replied [to the email], but it hadn't been successfully sent

It's the past-negative-potential-teiru form of 送る. This (て)いる refers to the continuation of state.

• 送る: to send
• 送れる: (potential) to be able to send
• 送れている: (potential-teiru) have been able to send
• 送れていない: (negative-potential-teiru) have not been able to send
• 送れていなかった: (past-negative-potential-teiru) had not been able to send

The sentence means "I thought I replied, but [I noticed] I hadn't sent [the mail] (successfully)".

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