No one actually "owes" any child a big, fancy wedding. It is traditional for the bride's parents to pay for a wedding, and traditionally, that wedding was planned by the parents with minimal input from the bride and none from the family of the groom except possible a guest list. The family planned a wedding that fit their finances, whether that was cake and punch or a six course meal and orchestra at the country club.
The bride was married from her parents' house, where she still lived, and moved to her husband's house.
These days, everyone thinks they need a huge affair (believe me, I was in the business for years) and the selfish ones think that their parents should mortgage the house to pay for it. More reasonable couples, especially couples who are established in jobs, usually pay part or all of the costs themselves. So the new rule is, if the parents are paying, the parents plan according to their ability to pay. If the couple is paying, they can have whatever they want and can finance.
As you're dealing with two sets of parents for the bride, generally the parent the child lives with plans and asks for a contribution from the other parent (often demanding money the other parent can't or won't pay.) But you don't have that problem, as her mother is willing to pay. In this case, her mother gets to do the planning. In order to keep family peace and be polite, she should tell you how many guests you are allowed to invite, ideally giving the groom's family half the spaces, and splitting the rest with you, though she may give you fewer if her family is larger, etc. If you need to invite more people than she says, you should offer to pay for dinner for the extra number you wish to invite. You should pay for your dress, your husband's tux, and whatever your children wear, even if they're in the wedding party. If you will travel to the wedding location, you should pay for your own hotel rooms and travel expenses. That's really all you need to do, but as a gesture of love and support for your daughter, you might offer to pay for a specific part of the wedding, such as the flowers (be careful, a rich mom might buy LOTS of flowers) or the champagne or the limo, if you're having one.
The thing that is NOT your responsibility is the shower. Traditionally, a shower is not given by an immediate family member, as that implies asking for gifts for your family. Showers are given by the bridesmaids, by a friend of the bride's or groom's mother, or possibly by an aunt. Showers are not properly given by mothers or sisters. However, many people now do it anyway, and if that's the custom among your friends, then giving her a shower is nice of you.
P.S. It would be nice if you wear whatever color the bride's mom asks you to wear and not fight with her about it.