Keep in mind synthetic rubber flooring for your laundry room. You simply can't beat the traction of rubber soled shoes on rubber flooring. Synthetic rubber flooring is fairly expensive, but it's extremely durable. It's the only flooring I know of that's used in golf shops and skating rinks where people will be walking on it with spiked shoes and skates on.
The biggest name in synthetic rubber flooring is Johnsonite. They have a very wide selection of synthetic rubber floor tiles for a wide variety of applications.
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Maybe take your old tiles off and just leave the floor bare until you can put down synthetic rubber floor tiles.