Check that all the custom chips in your 1000 are seated properly . It could be that it has some ROM which holds the kickstart and that could have become slightly unseated over time .
Just push them all down with a bit of force (not too much) till u hear them click into place .
Also check the floppy drive cables as they may have become loose over the years .
Apart from that i can't think of what may be wrong with your A1000 as it seems to basicly be working fine apart from that problem .
As others have said , also try your A500 floppy drive too if all else fails . And make sure your K/S disk is working properly . I dont think you can try it on your A500 as the A500's kickstart is already built into the ROM .