For John Sullivan M 1863 +/- 4 yrs with the father's forename unknown, I found 5 rcds: 1859, 1863, 1864, & 2-1865. You would have to view each detailed rcd to see if one (or more) show a spouse named Kate (most likely as Catherine).
For Cath(erine) Sullivan B ditto with a father's forename of John, I found 0 rcds.
For Mary Sullivan B ditto ditto, I found 0 rcds.
Co. Kildare rcds are generally good by this date. Am surprised I didn't find anything on the children's names; these are quite common.
Note - the surname Sullivan is heavily concentrated in Co. Cork and Kerry. Both of these counties likely have 1 million undigitized rcds