In PE22/152, Tolkien gives miriéla as the cognate of míriel. He rejected this word, but I think it suggest that we should derive this sense with an active participle like mirilya- → mirilyaila, *mir- → mirila tintila- → tintilaila, ita- → itaila...
Contribution “mírina” by Sámo Collarwa
These are the comments on Sámo Collarwa's contribution “mirilya”.
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Sámo Collarwa
#2382
I believe mirila might be best, as the first entry is perhaps a bit much on the tongue. itaila isn't bad, but it fails to attain a similar spelling to míriel. Would mirila maintain the same definition, or would it be changed?
Gilruin
#2403
Upon rethinking, I don’t think mirila works that well, it assumes a stem verb mir- from a non-verbal root MIR- “precious; esteem, value”. For your translation of A Elbereth Gilthoniel I would recommend mirilya as an adjective miril + -ya “shining-jewel-like”, which works differently from Sindarin’s míria-participle in terms of grammar, but has a similar outcome.