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.
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 ...
Here is another set of examples aimed at the precious bonus points.
Now the cases are in the order they are taught here (nom, acc, gen, dat, abl) so as to help memorization; feel free to permute to your local standards.
The first example uses only first declension feminines.
You can also switch to plural for those endings.
Puella uvam amicae vicinae e ...
Here is an example using all seven cases in a typical way:
Marce, vir feminae panem e furno pistoris Romae dat.
Marcus (voc.), the man (nom.) gives a bread (acc.) from the baker's (gen.) oven (abl.) to the woman (dat.) in Rome (loc.).
Reason for each case:
Marce, vocative: Marcus is being addressed ("Hey Marcus!"), and the vocative is used for this....
Here is an all-masculine attempt, one word per case, plus a verb:
Vesperi, Attice, imperator populi iussu regi equum pollicebitur.
In the evening, Atticus, the commander, on the people's order, will promise the king a horse.
Vesperi: Locative of vesper.
Attice: Vocative of Atticus, the person to whom the narration is addressed.
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: