The morphological issues are explained already. In any case, I hope that a literal translation will help:
"The day of wrath, that wellknown day"
About the use of illa, in this context, I would say that it is used to indicate some well-known or celebrated object, equivalent to the ancient, the wellknown, the famous.
You can find this use here:
It is the feminine nominative and refers to dies. It means “that day.”
You do not say why you think you can definitely rule it out, but I guess you think dies is masculine, which is indeed the case. But it is also often feminine. Generally speaking, it is feminine only when referring to a set day, an appointed time. This does arguably apply to the day of ...
This means "illa" definitely doesn't refer to "dies".
But it does!
The word dies can be feminine, and it is here.
The feminine gender is rarer but it is the typical choice for a special day like an appointment or a deadline.
That's why it was chosen here.
For details, see this question about gender variation in dies.