This OED webpage uses the word, 'secondly', at the bullet starting 'Anna Seward'.

Very good point, Ardiril.

oed.com: Passage following is for the second meaning.

2. In the second place; as the second in serial order. Chiefly in the enumeration of heads or topics in a discourse.
c1374 Chaucer Troilus & Criseyde ii. 1741 Secundelich þer yet deuyneth noon Vp-on yow two.
c1425 Lydgate Assembly of Gods 106 Furst, to begynne, thys Eolus hath [etc.]... Secundly,..Ofte of myn entent hath he made me mys.
c1450 R. Holland Bk. Howlat 352 Syne in a feild of siluer secoundlie he beris Ane Egill ardent of air.
1535 Bible (Coverdale) 2 Kings xvi. 19 Secondly, whom shulde I serue?
a1610 J. Healey tr. Epictetus Manuall (1636) 51 Man, consider first the nature of the thing that thou intendest, and secondly thine owne nature.
1692 R. L'Estrange Fables ccxiv. 185 We are Taught here Principally, Two Things; First,..And yet 2ly. How Prone we are to Indulge our Own Errors [etc.].
1712 H. Prideaux Direct. Church-wardens (ed. 4) 12, 1st, For absenting from Church. 2dly, For not abiding there till..Sermon be ended.
1863 C. Lyell Geol. Evid. Antiq. Man 36 Secondly, if the mud pierced through had been thrown down by the river in ancient channels, it would have been stratified.

How high do they go?

Fourthly?
Fifthly?
Twelfthly?

I was taught that when you're listing things you say "First" not "Firstly", but then "Secondly", etc.

We looked up fifthly last night on the Miriam Webster site - it's a word too. And as I was typing that, Firefox recognized it as a correctly spelled word. It does not like twelfthly though - or seventhly. So Firefox goes up to sixthly.