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while (<DATA>) {
   s/^"\d+"/'"' . ++$i . '"'/e;
...
     "time: 1600";
     "sump:  3";
}

perl -i.bak -pe 's/^"\d+"/++$i; qq{"$i"}/e;' file          <- Unix
perl -i.bak -pe "s/^"""\d+"""/++$i; qq{"""$i"""}/e;" file  <- Windows