Could he be a wali (guardian) to the daughter of his wife who was born as a result of an illegitimate relationship?

Firstly: 

A child born as a result of zina (illicit relationship) is
not to be named after the zaani (the man who committed zina), as previously
explained in the answer to question no.
117 

So this daughter is like one who has no relatives on the
father’s side and has no wali for marriage among her relatives, so her wali
is the Muslim ruler – if she is living in a Muslim country. This is
indicated by the report narrated by Abu Dawood (2083) and al-Tirmidhi
(1102), according to which the Prophet (blessings and peace of Allah be upon
him) said: “The ruler is the wali of the one who has no wali.” Classed as
saheeh by al-Albaani in Irwa’ al-Ghaleel, 1840 

If she is in a non-Muslim country, the director of the
Islamic Centre takes the place of the Muslim ruler, so he is her wali for
marriage. 

For more information, please see the answer to question no.
144712. 

Some scholars – such as Imam Abu Haneefah (may Allah have
mercy on him) – give precedence to relatives on the mother’s side over the
ruler with regard to guardianship, such as the maternal uncle or
grandfather, because they are closer and more compassionate than the ruler. 

As for the husband of the mother, he cannot be a wali for the
girl in marriage, because he is not one of her male relatives on the
father’s side or other relatives. 

Secondly: 

With regard to your saying: “I’ve heard the hadeeth about how
daughters will be a shield for their father from the Fire on the Day of
Judgement”, 

It was narrated from the Prophet (blessings and peace of
Allah be upon him) that children who die when young will intercede for their
parents on the Day of Resurrections. 

For more information please see the answer to question no.
159942. 

This daughter is not your child, so it is not permissible for
her to be named after you. 

But you will be rewarded, in sha Allah, for bringing her up,
spending on her and looking after her affairs. Allah, may He be glorified
and exalted, say (interpretation of the meaning):

“So whosoever does good equal to the weight of an atom (or
a small ant) shall see it.

8. And whosoever does evil equal to the weight of an atom
(or a small ant) shall see it”

[al-Zalzalah 99:7-8]. 

And Allah knows best.