SOURCE_OF_NAME:
The surname WOZNICKA was provided by family members, however, the death certificates for both Aniela and her sister, Pauline, list the father's name as KENSICKI.

BIRTH:
Date of birth given as July 20, 1854 and her father's name given as Joseph Kesicki on her death certificate.
Census listed her for Aug 1853.

DEATH:
From the Ohio Death Index (Note the mispelled surname):
Name: CZACHERSRKI ANNA I
County Name: CUYAHOGA
Date of Death: 2/6/1918
Volume Number: 2499 Certificate Number: 8434

1. Bertha WITWER, (Witfer) b. 3 Jun 1882, Poland; d. Feb 1958, Cleveland, Cuyahoga, Ohio, USA.

Woz~nicki is pronounced roughly "vozh-NEET-skee." It comes from the same basic root as Wozniak -- woz-, "convey, transport (especially by cart)." But generally would refer to the name of a place where the family lived at some point centuries ago. Thus we'd expect it to mean "one from Woz~niki or Woz~nica or Woz~nice." On modern maps I find one Woz~nice and a lot of Woz~niki's, so the surname would generally mean the family came from one of those villages; but without detailed genealogical research into a specific family's background, there's no way to know which one. The places, in turn, would take their names from some connection with carts, cart-drivers (woz~nica is a term meaning "cart-driver," and this name is an adjective that could come from that noun), cart-horses (woz~nik, and the name can also come from that), etc. As I say, there is a linguistic connection with Wozniak in that both come from that basic root meaning, but in practice there wouldn't necessarily be any blood connection between the two names. Woz~nicki is a name in its own right. 
--Source: http://www.polishroots.org/surnames/surnames_38.htm

In total, there were 12 children but we only know the names of two.

1. 1. Aniela (Anna) KENSICKI, (Kesicki) b. 20 Jul 1854, ,, Poland; d. 6 Feb 1918, Cleveland, Cuyahoga, Ohio, USA.
2. Pauline KENSICKI, (Kesicki) b. 20 Jun 1864, ,, Poland; d. 22 Aug 1950, Cleveland, Cuyahoga, Ohio, USA.

1. 3. (Female) PIOTROWSKI
2. Anthony PIOTROWSKI b. Abt 1855.