I really find it difficult to believe this is Mary's son given her age in 1857 (and the gap between Sarah's birth and his).  But then, whose son is he?
.  Henry Clay MIMMS, b. 13 Aug 1857

1.  Jordan R. Dodd, ed.  1993.  KY, NC, TN, VA, WV Marriages:  Early to 1850.  Liahona Research, Orem, UT (Broderbund CD-229):  not found.

2.  1830 Census Index and Digital Images (online at Ancestry.com; Image #57-58 of 98; extracted by Diana Gale Matthiesen):

1830

KY

Logan Co.

Russellville

pp. 65B-66A

John W. Mimms

201 110 000 - 000 100 100

200 000 - 220 100

These data indicate:

No., Race, Gender	Age Class	Therefore Born	Individuals Inferred
2 white males	4 or under	1825-1830	= Robert (b. 1830) = Son A
1 white male	10-14	1815-1820	= Brother
1 white male	15-19	1810-1815	= Brother David (b. ca. 1813)
1 white male	20-29	1800-1810	= John (b. 1805/6)
1 white female	15-19	1810-1815	= Mary (b. 1809)
1 white female	40-49	1780-1790	= Mother Lucy (POOR) MIMS
2 male slaves	9 or under	1820-1830	= ? = ?
2 female slaves	9 or under	1820-1830	= ? = ?
2 female slaves	10-23	1806-1820	= ? = ?
1 female slave	36-54	1775-1794	= ?

Both of Mary's parents were deceased by 1830, so the older woman is not Mary's mother.  John's father had died in 1828, and it appears his widowed mother, Lucy, was living with them, which also explains the older boys, who were John's younger brothers.

3.  1840 Census Index and Digital Images (online at Ancestry.com; Image #21-22 of 89; extracted by Diana Gale Matthiesen):

1840

KY

Logan Co.

pp. 189B-190A

John Mimms

122 001 - 100 010

020 000 - 020 100

0300000

These data indicate:

No. & Sex	Age Class	Therefore Born	Individuals Inferred
1 white male	4 or under	1835-1840	= David (b. 1835)
2 white males	5-9	1830-1835	= Drury (b. 1833) = John (b. 1831)
2 white males	10-14	1825-1830	= Robert (b. 1830) = Son A
1 white male	30-39	1800-1810	= John (b. 1805/6)
1 white female	4 or under	1835-1840	= Lucy (b. 1839)
1 white female	20-29	1810-1820	= Mary (b. 1809?)
2 male slaves	10-23	1816-1830	= ? = ?
2 female slave	10-23	1816-1830	= ? = ?
1 female slave	36-54	1785-1804	= ?
3	persons employed in agriculture

Where is Mary (b. 1837)?  Did she die before 1840?

4.  1850 Census Index/Images (online at Ancestry.com; Image #38 of 128):  District No. 2, Logan Co., KY, p. 90B-91A, 277/277, enumerated 30 Aug 1850, official enumeration date 1 Jun 1850 (extracted by Diana Gale Matthiesen):

John Mims	44	M	Farmer	1850	Virginia
Mary   "	40	F			"
John   "	18	M	Farmer		Kentucky
Drewry "	17	M	"		"
David  "	14	M			"
Lucy   "	10	F			"
Nancy  "	7	F			"
[next page]
Zerelda Mims	5	F			Kentucky
Thomas   "	4	M			"
Sarah    "	1	F			"

5.  1860 Census Images (online at Ancestry.com; Image #101 of 108):  Boarding House, Kansas City P.O., Kansas City Division 35, Jackson Co., MO, p. 101, PN 101, enumerated 18 Jun 1860 (extracted by Diana Gale Matthiesen):

837	886	Thomas M. James	37	M		Boarding House	10,000	1,500	Kentucky
		Sarah A.    "	33	F		House Keeper			Virginia
		John W.C.   "	11	M					Kentucky
		Luther T.   "	9	M					Do
	887	John W. Mimms	55	M			2,000	1,500	Virginia
		Mary      "	51	F		House Keeper			Kentucky
		Robert    "	30	M					Do
		Lucy F.   "	22	F					Do
		Nancy C.  "	18	F					Do
		Zeralda   "	14	F					Do
		Sarah     "	11	F					Do
		Thomas M. "	12	M					Do

Mary was born in Virginia (see 1850 census).  Thomas Martin JAMES is Mary's younger brother. 

6.  1870 Census Every-Name-Index/Images (online at Ancestry.com):  can't find, and I really looked.  Thomas Martin JAMES is still in Kansas City.

7.  Willard Rouse Jillson.  1925.  The Kentucky Land Grants. Filson Club Publ., Louisville, KY (online at Ancestry.com; Broderbund CD-650):

Volume I, Part 2, Chapter X, Grants in the County Court Orders (1836-1924)
Pg	Grantee	Acres	Book	Pg	Date	County	Watercourse
1496	Jno W Mimms	6	15	19	25 Apr 1839	Logan Co.	Red R