You can take your non-foldable bike only as far west as Leyton on the Central Line and only off peak. The map here shows where cycles can travel on the underground. The green lines are the permitted routes. Where a station is on the route it has access for cycles. The Central Line is a very deep line so once underground it wouldn't be possible to get a bike in and out of a station.
It is a 10km cycle from Leyton to Bank, use the Cycle Journey Planner to get the appropriate route.
Another option would be to cycle from Leyton to Maryland (~2km) and then catch a Greater Anglia service to Liverpool Street. Once again nonfolding bikes are restricted during peak hours. This is a National Rail overland service but a tube day pass and oyster card would be valid as long as the stations were in the purchased zones.
There are also plenty of overland options, it may be better to complete the whole journey via the overland if available. Unfortunately maps such as this one show all the overland lines, and is relatively simple to read, but it does not give any indication of geographical distance. Pick up an A to Z street directory (out dated these days!) or do some searches and use the Journey Planner around where you are staying or entering London to see if there are any better or closer options.