Is there any record of how old Kakashi actually was when he invented the Chidori? I know it was before Kakashi Gaiden when Obito died, so he must have been less than 10 right?
You are correct to say that Kakashi invented Chidori prior to Obito's death, but his age is never officially stated. However, there are some details from the Boruto series (take with a grain of salt) where we can deduce that Kakashi would have been roughly 11 or 12 years old.
Minato was needed on the front lines at the time, leaving Kakashi, recently promoted to jōnin at age 12[Boruto episode 36] in charge. [...] Shortly after entering Kusa, they encountered an Iwa-scout, Mahiru. Kakashi tried to eliminate him with his new jutsu, Chidori, but the attack speed left him vulnerable to counter-attack