Shard drop rate is 10% and 2 shards per drop in average in a Legendary Days event. So, it takes almost 500 fights to get her. You need to buy combativity.
50 1x, 5 10x or 1 50x fights, all give you 50 fights, so your chance remains the same in any case.
There is a perfume booster which gives you one more shard per drop up to 11 times, that is 55 less fights per perfume you use for this event . However each perfume costs the same as 55 combativity. So, using the booster you get the legendary girl sooner but it won't be considerably cheaper.