Filing status

Possibly,  To qualify as head of household (HOH) the taxpayer must meet all of these tests [IRC Sec. 2(b)]:

1) The taxpayer is not married at the end of the year. (Exception: Married taxpayers can qualify as HOH by meeting the tests for Considered unmarried.)

2)The taxpayer paid more than half the cost of keeping up his home.

3)The home was the principal residence for more than half the year of either of the following:

•The taxpayer’s qualifying child (see Qualifying Child). Exception: Do not count a qualifying child who is married unless the taxpayer can claim him as a dependent. (But, the married child can be counted if the only reason the taxpayer cannot claim him as a dependent is that the taxpayer can be claimed as a dependent on someone else’s return.)

•The taxpayer’s qualifying relative who is the taxpayer’s dependent (see Qualifying Relative). However, for HOH purposes, two types of qualifying relatives don’t count: (1) a person who is a qualifying relative only because he was a member of the taxpayer’s household the entire year, and (2) a person who is a qualifying relative only because of a multiple support agreement.

4)The taxpayer is a U.S. citizen or resident during the entire year.

 Caution: The preparer due diligence penalty of $540 (for 2020) applies with respect to eligibility to file a return as HOH [IRC Sec. 6695(g)]. To comply with the due diligence requirement, tax preparers must complete Form 8867 (Paid Preparer’s Due Diligence Checklist) for any return showing HOH filing status.

Temporary absences. A taxpayer and his qualifying person are considered to live together even if one or both of them are temporarily absent from the home due to special circumstances such as illness, education, business, vacation, military service, or detention in a juvenile facility. It must be reasonable to assume that the absent person will return to the home after the temporary absence. The taxpayer must continue to keep up the home during the absence.

Divorced parents. Even though a custodial parent has released the right to claim the exemption to the noncustodial parent (see Releasing the Exemption to the Noncustodial Parent), the custodial parent may still file as HOH.

No.   If you did not claim a qualified dependent then you cannot file as HOH.   And since you did not claim her as a dependent you cannot claim any education credits.

The fact that she earned more than $4300 does not matter if she was under 24 and was a full-time student.   You could have claimed her as a qualified child dependent.

WHO CAN I CLAIM AS A DEPENDENT?

You can claim a child, relative, friend, or fiancé (etc.) as a dependent on your 2021 taxes as long as they meet the following requirements:

Qualifying child

• They're related to you.
• They aren't claimed as a dependent by someone else.
• They're a U.S. citizen, resident alien, national, or a Canadian or Mexican resident.
• They aren’t filing a joint return with their spouse.
• They're under the age of 19 (or 24 for full-time students).
• No age limit for permanently and totally disabled children.
• They lived with you for more than half the year (exceptions apply).
• They didn't provide more than half of their own support for the year.

Qualifying relative

• They don't have to be related to you (despite the name).
• They aren't claimed as a dependent by someone else.
• They're a U.S. citizen, resident alien, national, or a Canadian or Mexican resident.
• They aren’t filing a joint return with their spouse.
• They lived with you the entire year (exceptions apply).
• They made less than $4,300 in 2021.
• You provided more than half of their financial support.

When you add someone as a dependent, we'll ask a series of questions to make sure you can claim them. There may be other tax benefits you can get when you claim a dependent.

From what you posted she is a qualifying child and you can file as Head of Household of Household even though she is not your dependent.