R u sure the sd card slot is broken? Could it be your sd card? Have u tried reformatting it as fat32 or using a different SD card?
If your sd card slot is truly broken, u can still mod your wii except for a couple steps that provide extra protection, but no functional difference so long as u don't do anything stupid. All you need to do is use modmii as per normal, but use a fat32 formatted hard drive plugged into usb port0 (the one nearest the edge) in place of an SD card, and when choosing an exploit use str2hax (doesn't require an SD card).
Without an SD card u can still do everything except install bootmii and make a nand backup.
All I have just said wil be stated in your custom modmii guide
Edit: when I say use a hard drive in place of an SD card, I mean follow the guide but every time it says sd card replace it with hard drive (I. E. Even the copy to SD folder should be copied to the usb hard drive)