If you look at the lower left portion of the above picture, this is the screen you'll use to influence other countries. To influence other countries, you'll need to use the Prio column (short for priority). Each day, you have a set amount of influence you can use, and this is where you choose which countries to use it on. Using the priority column, you can set an individual country to a priority of 1, 2, or 3. So if you have one country as 3 and another as 1, 75% of your influence will got to the first, 25% to the second. You can set influence to as many countries as you like, but it'll be a smaller piece of the total for each.
How influence works: There are several stages of influence: Neutral, Cordial, Friendly, Sphere etc. To move from one stage to the next, you need to build up 50 influence (to move from friendly to sphere you need 100. So once you have 50 influence on neutral, you can increase to cordial, 50 cordial to friendly, 100 friendly to sphere. You want to sphere all the German countries. Once you have the necessary influence, you should get a green circle on the diplomacy tab (on the main screen), and just click on Increase Opinion to upgrade. You should also familiarize yourself with Ban Embassy, Expel Advisors, and Discredit.
My recommendation for this:
1) Influence Holstein. Once this is done, you can attack Denmark for the state you need. With Holstein sphered they will stay neutral.
2) Move Hannover to cordial, then wait for Britain to de-puppet Hannover. There is an event for Britain that causes Hannover to no longer be a puppet. Once this happens, Britain won't actively prevent you from influencing it.
3) Start working on influencing Saxony. This will be the hardest of the 3 to take. The Austria AI will use influence to stop you, so you'll have to over-power them. The Austria AI will also defend Wurtenburg/Baden/Bavaria, so you can put a small amount into these and try to spread Austria too thin.
One strategy that works well is to put a single point into a country to don't want to sphere yet. The AI should instantly discredit you, which costs 25 influence. Then stop influencing there essentially wasting that 25 influence.
Some things you can do to make this easier:
1) Invest railroads/factories in the target country. Investing decreases the in luec down how er countries gain there, so you can slow down how effectively Austria can influence against you. Since you'll eventually be annexing all of these countries, there's not much cost in building railroads/factories there. Also don't invest in Bavaria as Bavaria has a small chance to become a Great Power which can be annoying.
2) Increase your relations to 200 with target countries.
3) The 4th column in the Economic tech will increase how much you can influence (the total number that is then split to the countries you're influencing).
4) Increase your overall score. The higher your score, the more influence you can gain.
5) Austria doesn't border Baden, and you do, so you'll have the advantage over them with Baden.
It is possible to do this in 4-5 years without warring Austria. It just takes some luck and some practice.