"It is better with a private health insurance."
Translation:Mit einer privaten Krankenversicherung ist es besser.
Duolingo's sentence is fine; your suggestion is not correct. I think you're mixing up article words (a, the, my,...) and adjectives (nice, small, private,...). They have different endings.
"Mit einer privaten Krankenversicherung":
einer: dative, feminine [dative because "mit" is always followed by the dative case; feminine because "Krankenversicherung" is feminine]
privaten: used after ein-word, dative, feminine [dative because of "mit", feminine because "Krankenversicherung" is feminine]
Article endings: http://german.about.com/library/blcase_sum.htm
Adjective endings: http://coerll.utexas.edu/gg/gr/index.html?si=adj
EDIT: At some point now long in the past, I incorrectly stated:
"Just to add to what Katherle said -- "einer" is not an article here, it is a possessive adjective. See here for a chart that includes possessive articles:"
EDIT: the chart is still useful though, even though I was wrong: http://www.lsa.umich.edu/german/hmr/grammatik/basic_chart.html