[Contains parts of a pretty definitive explanation from another blog (Neko to Negima! to Seiyuu-san)]
In the Shinto ritual prayer Minadzukitsugomori-no-Ooharae, Ibuki-donushi is one of the four gods/’pillars’ of Haraedo-no-Ookami, the God of harae, or purification rituals. The role of Ibuki-donushi is ‘blow away’ the sins of this world to Ne-no-kuni/Soko-no-kuni, one of the underworlds comparable to the more well-known Yomi-no-kuni.
The definition of ‘sin’ was not limited to morality but extended to ‘calamities’ and ‘impurities’*. In terms of Konoka’s usage, ‘sickness’ itself is a form of ‘calamity’, and thus falls under the domain of Ibuki-donushi.
* If you’ve watched or read, say, Kannagi, this should be familiar (kegare).
Ibuki-donushi was also attributed as Kamunahobi-no-kami; nahobi means more or less ‘restoration’, and its verb form nahosu is the origin of the words for ‘straighten’ and ‘treat/heal’ (naosu). Therefore, restoration was also one of the domains of Ibuki-donushi.
“A purification ritual to Ibuki-donushi”
As noted above, harae is a purification ritual – ooharae is literally a great purification, for the benefit of an entire community or even the whole country, usually a public event.
Takamagahara ni kamudzumarimasu Kamuroki Kamuromi no Mikoto wo mochite
“I enter the High Plains of Heaven invoking the power of the gods”
Takamagahara is the seat of the Gods. Kamuroki-no-Mikoto and Kamuromi-no-Mikoto are the Gods of male and female, respectively.
Sumegamitachi no mae ni mausaku kurushimi urefu waga tomo wo mamori meguma hi saki hahe tamae to
“Humbly gathered before the Imperial Gods, my suffering comrades, protect them, bestow upon them blessings and good fortune”
Fujihara no asomi Konoe Konoka no
“I, Konoka of the Konoe house of the Fujihara clan”
The Konoe house was (or is, considering it still exists) indeed one of the five regent houses of the great Fujihara clan. Konoka is referring to her ‘full name’ as per ancient tradition, hence the deliberately rough rendition: Fujihara, her blood; Konoe, her origin; and Konoka, her personal name.
Asomi is a hereditary title of rank and political standing, attached to the clan in one’s name as seen in Konoka’s example. It was one of the most prestigious (and politically powerful) titles.
iku musubi wo udzu no mitegura ni sono he tatematsuru koto wo moromoro kikoshimese
“Offer my soul within this cloth as a treasure to the Gods so They may heed my humble request”
Or to put it more directly, offering her life. Mitegura refers to offerings to the gods or Gods during rituals (at least consisting of food and alcohol). This will generally be wrapped in a cloth, hence the image is of Konoka being wrapped as mitegura for the Gods.