"Doe of the Dawn" appears to be an accurate translation. אַיָּלָה (ʾayyālāh) means "doe". (Morphologically, it is the feminine of אַיָּל, meaning "deer.") The word in question, אַיֶּ֥לֶת (ʾayyelet), is the construct form: "doe of...". The following word שַׁ֫חַר (šaḥar) is a common word for "dawn." It is prefixed with the definite article making the whole construction definite: "the doe of the dawn."
The reason the phrase is "missing" from the KJV at the linked site is that it is part of a superscription, a heading present in the text of 116 of our 150 Psalms. These are not included in the numbering of most English versions but constitute verse 1 in the Hebrew (and usually the Greek). Although missing from the link in question, these were in fact included in the KJV, including the superscript above Psalm 22.
The KJV renders a transliteration only, not venturing a translation1 presumably due to uncertainty about the significance of the phrase, although the words themselves are straightforward (but see note). As with most of the superscriptions, there is an element of uncertainty about the original intention. They are most often considered liturgical notes, and the על (ʿal; "according to") phrases such as this one are often taken as references to a tune to which the Psalm was to be sung.2
1. Except in a footnote: "or, 'the hind of the morning'", as helpfully pointed out in a comment.
2. John Goldingay, Psalms 1–41 in Baker Commentary on the Old Testament Wisdom and Psalms, ed. Tremper Longman III (Baker Academic, 2006), 324. Goldingay takes a different understanding of the phrase, translating "dawn help", following the LXX and a less common meaning of ʾayyelet (precedented only in v. 20 ʾĕyālûtı̂, [ESV: "my help"], which is otherwise a hapax legomenon). This phrase he takes as "presumably a tune or a way of singing."