The “Sick Man of Europe” originally described the Ottoman Empire (Turkey).
 
 
Wikipedia: Sick man of Asia
The phrase “sick man of Asia” or “sick man of East Asia” (Chinese Simplified: 东亚病夫, dong ya bing fu) originally referred to China in the late 19th and early 20th centuries when it was driven by internal divisions and taken advantage of by the great powers. China was forced to sign a series of unequal treaties, culminating in the Japanese invasion of China during World War II. The phrase may be considered derogatory, as it mocks Chinese government and the Chinese people for being weak. The phrase was intended as a parallel to “sick man of Europe”, referring to the weakening Ottoman Empire during the same period. Like the “sick man of Europe” term, it has also been used to refer to numerous other Asian countries.
 
(Oxford English Dictionary)
sick man  n. a term frequently applied, during the latter part of the 19th cent., to the Sultan of Turkey. Also figurative, originally applied to Turkey and hence to other countries, regions, etc., and in extended uses.
Quot. 1853   refers to a conversation between the Tsar Nicholas I and Sir G. Seymour at St. Petersburg on the 21 Feb. 1853.
1853   Ann. Register, Hist. 252   I am not so eager about what shall be done when the sick man dies, as I am to determine with England what shall not be done upon that event taking place.
1855   J. Martineau Ess., Rev., & Addr. I. (1890) 428   It was all right not to let the ‘sick man’ be frightened into convulsions.
1860   J. L. Motley Hist. United Netherlands I. ii. 30   That formidable potentate, not then the ‘sick man’ whose precarious condition and territorial inheritance cause so much anxiety in modern days.
1860   S. S. Cox Eight Years in Congr. (1865) 129   ‘Mexico is our “sick man”.’ ‘Yes; she is to America what Turkey is to Europe.’
         
OCLC WorldCat record
Covering the Far East
Author: Miles Walter Vaughn
Publisher: New York, Covici, Friede [©1936]
Edition/Format:   Print book : English
Contents:
Sick man of the east—
 
OCLC WorldCat record 
Philippines - Sick Man of Asia no More
Author: R Subbaraman
Edition/Format: Article Article : English
Publication: CURRENT ECONOMICS -JOURNAL OF INTERNATIONAL ECONOMIC RESEARCH- (January 1999): 17-19
 
OCLC WorldCat record
The sick man of Asia : China’s health crisis
Author: Yanzhong Huang
Publisher: New York : Council on Foreign Relation, 2011.
Edition/Format:   Print book : English
Publication: 2011, Vol.90, No.6 (p.119-136) Foreign Affairs .
 
OCLC WorldCat record
Revisiting the “Sick Man of Asia” Discourses of Weakness in late 19th and early 20th Century China
Author: Li Fan; Clemens Büttner; Zhang Ke; Joachim Kurtz; Li Lingfeng; All authors
Publisher: Frankfurt Campus 2018
Series: Schwächediskurse und Ressourcenregime Discourses of Weakness and Ressource Regim, 2
Edition/Format:   Print book : English : 1. Auflage