How can a flip flop alternate between two clocks, turning one off and the other on and repeating?
Probably, the most simple way to produce the clocking signal you need is to produce a single clock \$Q\$ with a period of 20ms and a 90% duty cycle: by inverting such clock (for example by using a NOT gate or a single fast transistor inverter) you will get another clock  \$\overline{Q}\$ with the same period but with a 10% duty cycle. Each of this waveform is shown below:



Then, the original clock signal will provide the 18ms pulse, while the inverted one will provide the 2ms, for example by ORing the outputs of the following simple RCD trigger circuit:







simulate this circuit &ndash; Schematic created using CircuitLab

By calling \$t_1\$ the trigger generated by feeding \$Q\$ to the input the of the above circuit and \$t_2\$ the one generated by \$\overline{Q}\$, these signals have waveforms similar to the following ones:



And ORing the two triggers you'll get the following composite one:



A few notes


You can use one 555 to generate \$Q\$ and then inverting its output: however the simplest way is perhaps to generate both \$Q\$ and \$\overline{Q}\$ by a single, properly designed, astable multivibrator.
The RCD trigger circuit show above is simply an analog differentiator: \$R\$ and \$C\$ control the duration of the trigger pulse, while \$D\$ rules out the negative pulse which would otherwise appear during the falling of the input squared wave signal.
This is an hybrid analog/digital solution to your problem: if you need to work with only logic gates, perhaps the solution to which awjlogan alludes is the best.