A few people within this thread have mentioned dance. I can say that personally, I've experienced this working! About five years ago, I took up ballet, jazz, tap, modern and ballroom dance. I (and others) began to notice my Aikido steadily improving. Now I am a swing dance instructor. Swing dance has great exercises for partner work, I recomend it.
As for Aikido practice by yourself, I (like a few other people have done) recommend practicing basic movements that later become components of your techniques.
Try some of the following:
Irimi - sliding entering movement
Tenkan - two steps (one foreward, and one backwards = turning you 1880 degrees)
Tenshin - one step off the line, step backward at a 45 degree angle with the other foot
There are many other things you can do, such as independant (warm-up) exercises, like sai undou and irimi undou, if those are things that your dojo practices.
Also, don't forget that you can 'run-through' techniques without a partner. In my opinion, this is a rather advanced concept and difficult thing to do. It's good practice. If you do any weapons training, you can find a great rhythm in that. If you have a partner to practice with sometimes, you can do the blending timing of a shomen-uchi (head strike) attack multiple times, and the timing will come.
I was speaking with one of my Sensei at Hombu Dojo just the other day about how difficult blending is, and how timing is the hardest thing for a lot of people. It comes with time and practice.