Chelsea getting a huge cash injection is not going to "solve" anything for them. Lets be real, they do not have a bad team anyway, and they got a good manager, that is not the problem and money certainly isn't because they have some awesome players they bought as well, overpaid mostly too.
The issue they have is that they are not a "team", like if anyone thinks that City has the best team in the world, or at least in this league, has no idea about football, they have one of the best, but not better than everyone enough to win it four in a row. Why are they dominating then? Because they are a team, and being a team beats anything else in the world. Chelsea needs to learn to be a team, and if they do that, then sky is the limit for them, they have everything else ready.
It's not all about money but the Premier League is also the toughest competition in the world. Management governance and ways of making the right decisions will be one step ahead compared to other teams. Chelsea need to learn a lot and they are not only able to hunt for players, but the management side must also be filled with talented professionals. So it's useless if you have enough money but are always careless in making decisions and that won't make a good change. Manchester City is currently one of the teams that has a pretty good management structure in the Premier League so they are always able to see the situation and make very careful decisions.