Learn English – Why is this wrong? I have become sick for 2 days

Yes, has been around means 'has been in existence'. The OED gives only ‘to have gained much experience in the world’ but in practice the expression is used rather more widely, if only in informal contexts.

Green's Dictonary of Slang identifies two senses of "for days," the first is an exclamation that originated in the U.S. LGBT community.

the orig[inal] implication was having sex continually, for day after day

1. (gay) an excl[amation] implying shock or amazement.

The one citation under this sense is dated 1970.

The questionnaire was a list of terms known to the investigator and thought to have the greatest currency: trick, basket, box, camp, queen, Mercy! For days!

• 1970 - J.P. Stanley ‘Homosexual Sl.’ in American Speech XLV:1/2 45

The second sense seems to more closely apply to the description in the question referring to "a lot of" something. This sense is described by GDoS as hailing from U.S. black culture as early as 1968 and with an attestation from 2002.

(orig. US black, also days!) a general intensifier implying an extreme, for a very long time, absolutely truthfully.

The 2002 usage is typical:

Six foot one with tight cornrows and curves for days.

• 2002 - ‘Touré’ Portable Promised Land (ms.) 185

Although GDoS treats these as two separate senses of the phrase with two separate cultural contexts, I think it is worth noting that there tends to be significant cultural exchange between the two communities, and I'm inclined to speculate that the term originated in one culture and drifted to the other. My interpretation of Green's entries and the note on the "original implication" is that he views the phrase as firmly rooted in LGBT culture.