R, 83/72 bytes

Alt approach using regexps

If we're allowed a trailing newline, 72 bytes:

function(s,n)cat(gsub("([^
]*
)",r,gsub("([^
])",r<-strrep("\\1",n),s)))

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Otherwise, 83 bytes:

function(s,n)write(rep(el(strsplit(gsub("([^\n])",strrep("\\1",n),s),"\n")),e=n),1)

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