This is a fantastic card. There are a decent number of effects which let you search (some of) your deck for a card and add it to your hand - specifically they let you look at 5 cards, 10 cards, or your entire deck. However the majority of such effects, including all the full deck searches, are one shot effects rather than being repeatable like the Mirror. Most of them are also more limited, only allowing you to search for a certain type of card. And if they're not, they're likely more expensive. Let's make a few comparisons:

• Gather Information, Heed the Dream, Word of Command and The Seeing-stone can search your entire deck, but one costs a round of questing, the next costs 4 resources for it, the next exhausts an Istari and the last only searches for Doomed cards; plus they're all one shot effects.
• Mustering the Rohirrim and Westfold Horse-breeder search 10 cards, but only specifically for Rohan allies and Mount attachments respectively, and once again they're one shot effects.
• Master of the Forge is repeatable every round, but costs one more resource, only searches for attachments, and only searches 5 cards rather than 10.

Meanwhile the Mirror searches 10 cards, is repeatable every round, and searches for anything you want. With advantages like that, it's little wonder they had to give it a downside in the subsequent random discard from hand. But that can be worked around in a few different ways:

• The first point is obviously that if your deck has a decent amount of regular card draw you may find yourself with a lot of cards in hand, which obviously gives you better odds of discarding something you can do without and keeping the more important cards you want to hold on to. Since the Mirror requires you to have Galadriel, that's a bit of card draw you'll automatically have, and Nenya then gives you access to all the card draw in the sphere, so your chances are reasonable.
• One aspect is simply to consider how you're using the Mirror. While it's tempting to use it to grab useful cards, an interesting approach you can consider is if your deck is efficiently playing its important cards, then you could use the Mirror to grab all the less useful ones you don't mind discarding, like duplicate uniques. That way the random discard isn't such a big problem and by removing those cards you're thinning out your deck so your regular draws will have a better chance of being useful.
• A counterpoint or corollary to that first case is that if you're playing a deck which primarily depends on a particular combo of cards, the Mirror can be very useful to find the combo pieces, and presumably everything not part of the combo is considered less significant and thus you won't mind so much if it gets randomly discarded.
• A deck with a lot of recursion may care a lot less about the random discard - effects like Dwarven Tomb, Stand and Fight and Caldara mean that a lot of the cards you might end up discarding you can just get back from your discard pile and tht may well be an easier way of getting them into play than waiting to draw and play them normally.
• Some cards you specifically want to get into the discard pile in certain decks. Allies for Caldara/Stand and Fight come up again here; but there are also Lords of the Eldar, Elven-light and Glorfindel, which play from the discard pile; and cards which benefit from having copies in the discard pile - Anchor Watch, Skyward Volley, The Evening Star, Elwing's Flight and Veteran Sword-elf. In a Caldara deck in particular, I've actually used the Mirror with an empty hand just to move an ally from my deck to my discard pile.
• Some decks may be designed on the principle of having lots of cards which are useful, but none which are absolutely essential so no discard is too big of a wrench - Erestor would be an obvious case for this since his ability means a lot of the cards can get discarded anyway.
• Finally, of course, the simplest option is just to play a Silver Harp, so after the random discard happens you can exhaust the Harp to return the card to your hand if it's something you want to hold on to. If you're really depending on this to dig for important cards, including the Harp does take away one of the advantages of the Mirror, that being its cheap cost of 1, but on the other hand having a repeatable 10 card search is absolutely worth 3 reources.

The Mirror is certainly not essential if you're using Galadriel - she provides her own card draw already and there are plenty of other valid draw options. Plus, the random discard may be considered too big a risk for some decks which may want to hold particular cards for a while, waiting for the right time ( staple A Test of Will is of course the obvious example of such a card), and for any deck which is designed to play efficiently and consistently play everything in its hand the Mirror would most likely be useless for a majority of the game. But there is a wide selection of decks for which the Mirror can work perfectly, and it's an incredibly powerful effect. With or without a Silver Harp to negate the downside, repeatably searching the top 10 cards of your deck for any card you want offers a level of flexible power which is hard to overstate.