If ibuprofen is no longer working to reduce the pain from a toothache, would dihydrocodeine be a good alternative?

Depends: Without a complete medical history and a diagnosis of what is causing the dental pain, you are self prescribing. Not a wise thing to do without knowledge of drug interactions and a diagnosis as to the cause of the pain.

Not for long-term: What do you propose to take when the Codeine does not work? This is a losing battle you are fighting. The pain is caused by a problem. The best way to make the pain disappear is to address the problem by seeing a dentist. I realize that it is sometimes difficult to visit a dentist, but if you don't address your dental problem, it will likely not resolve itself and will likely become worse.