1998 Oldsmobile Intrigue won't start

I don't know why your speedometer acts-up? I'd have to make sure the battery has a full charge and battery connections, coming and going are clean and tight, all ends. Next, I'd check your starter circuit. I haven't looked at your security system?

Check applicable fuses for starter circuit. Also, check for voltage at the fuse, use voltage tester. One is hot all the time, one goes hot with the key on, the other is hot with the key in the crank position. Any testing at the crank relay, make sure the tranny is in park or neutral and the e-brake is set.