No relation. I first looked to the census substitutes for Ireland. Griffiths Valuation 1853. The Tithe's were done in 1824 but know of no complete records.
I found 227 Griffith Records for Mitchell, however, for some odd reason a bmd data base included the name MOORE in their reporting methods.
There were commonalities between Mitchell's, Orford, and as expected Ryan's and Byrne.
I found no David Mitchell except for 1870 nothing earlier, doesn't mean there wasn't any.
The most common Parish they had in common was Kildrought Parish, Celbridge Townland, and Parish of Rathangan, but not a common Townland. The Parish church is where most people probably met unless they were Townies.
You can find the Griffiths at www.failteromhat.com
I just going to give a few eg.
1846 John Mitchell was listed as a Miller, Leixlip Kildare
Patrick and John Property #29, Kildrought, Celbridge
house&garden 0Acres0Roods0perches (old english measurement)
James Mitchell, Rathangan #3a house,office,land 2.2.35
Jas. & Thomas Mitchell, Rathangan #5a house,off,land 31.
A bmd database for Kildare, shows Patrick's born with a father named James: 1829, 1846, 1851, 1865
James Mitchell born 1796 and 1814, Anne Byrne's born 1795, 1805, 1806, 1811. Marriages of Anne Byrne 1829 and James 1829.
Ally Michel born 1830, 1 Mary Michel 1830, 1 Mary Mitchell born 1831 These could be the same Mary's, transcription error or one birth and then one baptism which might be two separate dates, generally a baptism took place the first Sunday after birth depending on circumstances. The website of this source cannot be put on rootsweb message board but I can give them to you in an e-mail: email@example.com
+ County are two generally good resourceswww.familysearch.org
PILOT found under 'search records'
David Mitchell died July-Sept, Naas Registration District Kildare/Wicklow Co.'s age 73 est birth year 1830 #101602
Vol.2, Pg. 559 (you should be able to get any records after 1864 from the Dublin General Registration Office: www.groireland.ie
but only the person reporting death is usually given (son,wife,daughter)or visit a Latter Day Saints History Centre to see the miscro film.
I didn't find Ally Mitchell which I thought would be the easiest, I will leave the rest to you
David Mitchell marriage 1870 Naas and birth 1901 see records
'genealogy for fun'