Can't offer too much input here other than I had the floor of my shower crack in an apartment several years back. The landlord paid to fix it, and never even so much as disputed it with me. In that case the damage was the result of him having installed it improperly (not enough support underneath). If I were you I'd be fighting this all the way, there is no reason for you to pay for this as it is definitely their responsibility unless you broke it while juggling bowling balls in the shower.
As everyone else is saying, this definitely doesn't sound like your fault / responsibility. As a former landlord, I would have paid for the repairs (and apologized to the tenants for any inconvenience) if this had happened. Are they telling you that you need to pay the repair people yourself or that they'll pay then pass the bill on to you? As Rachelle suggests, I'd try to move up the ranks and talk to supervisors (go as high as you can), and tell them that you'll have to involve legal aid or whatever your local version of the Landlord and Tenant Board is if they refuse to pay / make you pay for it / refuse to make repairs.
As has been suggested, keep paying your rent. A dispute with the landlord isn't justification to withhold rent.
Lots of good advice ha already been offered but I would just like to add that as a landlord when I had just paid for a brand new bathroom reno my tenant dropped his hair-product and cracked the sink and I paid for the repair, he offered to pay. I know many landlords wouldn't agree but I see these things as A) life B) tax deductable and C)more likely to keep my amazing tenants in place.
I think they should pay for the bath.
Now the cigarette burn in my bathtub....that's another story!
Is it just me who thinks that bathroom sinks and tubs that cannot handle something dropping on them ar very badly designed for a rental property? And life in general.
Just like the damn front loader Haier washer/dryer that cost a fortune, takes overnight to dry a load, and does not have a method of dealing with change and other various pocket stuff. The owner paid $500 to change the pump because of a bobby pin. The filter also is exactly the right size to hold a loonie or toonie right over the pump intake. Yes, people should empty their pockets but appliance manufacturers should design their machines to prevent machine failure cause by items commonly found people's pockets.