I'm not finding the sources that give Louis Is unknown mistress the name (or apparent family) of Theodelinde of Sens. The wikipedia citation has never been added.
At the moment her relationships are these:
Daughter of Gainfroi, count of Sens and Theidlindis de Blois
Wife of Reginmund van Friesland
Partner of Louis I, The Pious
Mother of Gisela of Lotharingia, Abbess of Nivelles Abbess of Nivelles; Arnoul and Alpaïs, Abbess of St-Pierre & Reims
These are the only sources, so it should be this:
Emperor Louis I had [two] illegitimate children by Mistress (1):
10. [ALPAIS ([793/94]-23 Jul 852 or after, bur [Reims]). Flodoard refers to "Ludowicus Alpheidi filie sue uxori Begonis comitis"[231]. The Annales Hildesheimenses name "filiam imperatoris…Elpheid" as the wife of "Bicgo de amici regis" when recording the death of her husband[232]. Settipani discusses the debate about the paternity of Alpais, preferring the theory that Emperor Charles I was her father[233]. If Emperor Louis was her father, it is unlikely that she was born before [793/94], given his known birth date in 778. It would therefore be chronologically tight for her to have had [three] children by her husband before his death in 816. However, no indication has been found in primary sources of the age of these children when their father died. The question of Alpais's paternity is obviously not beyond doubt, but it is felt preferable to show her as the possible daughter of Emperor Louis in view of the clear statement in Flodoard. No indication has been found of the name of Alpais's mother. If Alpais was the daughter of Emperor Louis, it is likely that she was not her husband's only wife in view of Bego's estimated birth date. After her husband died, she became abbess of Saint-Pierre-le-Bas at Reims in [817]. She was still there 29 May 852. m ([806]) [as his second wife,] BEGO, son of --- (-28 Oct 816). He governed the county of Toulouse as "marchio" for Septimania in 806. Comte de Paris [815].]
11. ARNOUL ([794]-after [Mar/Apr] 841). The Chronicon Moissacense names "quartum…filium [Ludovici]…ex concubina…Arnulfum" recording that his father gave him the county of Sens[234]. Comte de Sens 817. He was a supporter of his half-brother Emperor Lothaire in [Mar/Apr] 841[235].