What would we do instead? Keep Barack Obama?
There is zero provision in the United States for delaying the inauguration. In fact, it's not even clear that the inauguration is legally necessary. At noon on January 20th, 2017, Obama is out and a new president replaces him. There's a ceremony, but the constitution does not include any mention of it. The actual wording indicates that the new president takes office at noon, period. There is also some provision for what to do if the president-elect can't take office, but none of that delays the transfer of power.
It would have been possible (too late now) for people to have challenged the votes. This could have been done prior to the electoral college vote in December (and in fact there were recounts requested and at least one completed). Also, Congress could have refused to certify the votes when they counted them. Some people actually did attempt to refuse certification, but there wasn't any significant support for it.
It's also worth noting that Republicans had the votes to control the backup plan. If for some reason, no candidate received a legitimate majority of the electoral college votes, the vote would have gone to the House. The Republicans control more than the necessary twenty-six state delegations in the House. And the next step after that is the Senate, where Republicans control fifty-two of a hundred seats.
There are other questions about how this could have gone differently, but it didn't. And Republicans would have won regardless.
If we know for a fact that Russians interfered with the election, why can we proceed with inaugurating Trump without fully knowing the extent to which the hacks influenced the election results?
I find this question a bit confusing. Are you claiming that the Russian hacks influenced the election in and of themselves? We most definitely do not know that. The claim is that leaking the results of the hacks caused the election to shift from Hillary Clinton to Donald Trump. That remains contested.
What is generally accepted is that the hacks came from the Russians. Although the evidence for that is classified and not subject to public analysis. Also, additional classified, not public, information is said to prove that the Russians leaked the information to Wikileaks.
There remains zero evidence that the leaks of the hacked information impacted the election. Other than a few anecdotal stories, there is no proof that anyone chose to vote for Donald Trump and/or not vote for Hillary Clinton based on the leaked information. Even if there was evidence of that, it's not clear that that could invalidate election results. It is unquestioned that the relevant votes were cast by American citizens in accordance with their wishes at that time.
There is not now and never has been any proof that the Russians changed any vote totals. In fact, there is rather substantial proof that they did not. For example, the recount in Michigan did not indicate any signs of invalid votes cast nor significant differences between the local vote counts and the announced vote counts. Beyond that, it's not clear that they could impact vote totals via hacking. Election machines are not on the internet for exactly that reason. You don't hack them remotely but by gaining physical access and programming each individually.
Personally, I find the "Russians interfered with the election" construction misleading. What they are claiming is that the Russians made information available that was not public knowledge. They have no way of knowing what impact that had. However, even if it had no impact, they can still claim interference (however ineffectual). This lends itself well to political grandstanding but not to actual legal action to invalidate votes.