Dart solution

Here’s my solution to start the ball rolling. In the end I incorporated my part 1 answer into the part 2 logic, and golfed the code quite a lot.

import 'package:collection/collection.dart';

var ds = '0123456789'.split('');
var wds = 'one two three four five six seven eight nine'.split(' ');

int s2d(String s) => s.length == 1 ? int.parse(s) : wds.indexOf(s) + 1;

int value(String s, List digits) {
  var firsts = {for (var e in digits) s.indexOf(e): e}..remove(-1);
  var lasts = {for (var e in digits) s.lastIndexOf(e): e}..remove(-1);
  return s2d(firsts[firsts.keys.min]) * 10 + s2d(lasts[lasts.keys.max]);
}

part1(List lines) => lines.map((e) => value(e, ds)).sum;

part2(List lines) => lines.map((e) => value(e, ds + wds)).sum;