First let me give you the good news. Consistent, steady-state running works as well as speed work to improve your speed, at least at the start, and for runners at our age. Now, I'll ask a rhetorical question: Why do you think you need speed work? If you are looking for a change in your training, you probably don't need as much speed work as you suspect. However, it's a nice change of pace (no pun intended) from doing the same route at the same pace day after day.
If you're looking for a change of training focus, then you can do a number of good speed workouts without depending on a track. All these assume an out/back route, but you can adapt for a loop course:
Progressive Tempo Run - run at an easy pace for the first half (time), then slowly increase the pace for the same period of time for the second half.
Intervals - run easy for the first half, then pick up the pace for 30sec to 1min, with equal-to-twice the time back at the easy pace for recovery; if you pick up for 30 seconds, then run easy for a minute, pick up for a minute, run easy for two... This is kind of like fartlek (speed play) workouts, but a little more structured.
Your jumps in performance are probably due to the patient manner by which you have progressed so far. You won’t be hurt by continuing what you're doing for two to three more weeks before trying a 5K. Once you've done that you might think about either another three-to-four weeks of steady runs, or adding in one speed workout a week; do the progressive tempo one week, do the fartlek-type the next. If you plan to focus on speed for a specific distance, I would use both of those workouts two times a week, or split it up with a weekly tempo run (75% effort for 20-30 minutes). Add one long day (no more than 1/5th of your total mileage for the week) and easy runs during the other two or three days during the week. Oh, and one rest day a week is a must!
Make certain your form is good throughout your workouts; erect posture, “plumb line” from the top of head to the shoulders, spine, and hips; relaxed shoulders and hands, with your elbows bent at no less than a 90-degree angle. You should propel yourself across the running surface as smoothly as possible, with no “bounce” of the horizon, and as quiet a foot strike as possible. If you place an earplug in one ear you can listen to how hard you are pounding on the running surface. When talking about distance running, quieter is always better and more efficient.
After three-or-four weeks of that, then consider another 5K as a test to see how you're doing. After a second 5K test you might want to add an additional speed workout during the week. Let your conscience be your guide. As my coach has told me in the past, running is a sport of PACE and PAtienCE.
I had, however, an unusual phone conversation the afternoon after I got home. My ex-wife decided to call me after a 14-year hiatus; nothing better to do than to say hello and find out how my life was progressing. Before you think this was a less-than-civil conversation, let me stress that the last phone call we had, around Christmas of 1995, was, er, interesting. Interesting, in this case meant: 'boy was I a fool to run off rather than work out the marriage.' Sometimes the old phrase time heals all wounds is absolutely true. I was glad to hear her life had taken a turn for the positive; she had married a good man in her home town, earned her GED & was working for a major banking firm in the southeast US.
While I told her a lot about what I was doing here, I don't think she understood much of it. I definitely developed a different life in the 17 years since we divorced, one even more foreign to her than the one I had when we were married. As I hung up the phone, I realized something: Our intersection, while brief, was an important one in my education about life, women, leadership, and more. She's on her path, I'm on mine, and I hope she finds her happiness during the journey.